


















































































WORKS OF 

PROF. WALTER LORING WEBB 


PUBLISHED BY 

JOHN WILEY & SONS. 


Railroad Construction.—Theory and Practice. 

A Text-book for the Use of Students in Colleges 
and Technical Schools. Third Edition. Revised 
and Enlarged. i6mo. xvi + 727 pages and 232 
figures and plates. Morocco, $5.00. 

Problems in the Use and Adjustment of Engineer¬ 
ing instruments. 

Forms for Field-notes; General Instructions for 
Extended Students’ Surveys. Fifth Edition, Revised 
and Enlarged. i6mo. Morocco, #1.25. 

The Economics of Railroad Construction. 

T.arge i2ino, viii-j-339 pages, 34 figures. Cloth, 
$2.50. 




PROBLEMS 


IN THE 

USE AND ADJUSTMENT 

OF 

ENGINEERING INSTRUMENTS. 


Forms for Field-Notes. 


GENERAL INSTRUCTIONS FOR EXTENDED 
STUDENTS’ SURVEYS. 


BY 


WALTER LORING WEBB, C.E., 

Member American Society of Civil Engineers; Member American Railway 
Engineering and Maintenance of Way Association; 

Assistant Professor of Civil Engineering (Railroad Engineering) in the 
University of Pennsylvania, 1893-1901; etc. 


FIFTH EDITION, REVISED AND ENLARGED. 

FIRST THOUSAND. 


NEW YORK: 

JOHN WILEY & SONS. 
London : CHAPMAN & HALL, Limited. 
* 1907 . 


"PA56 2 - 

,VJ3(o2> 

UBftMY of CONGRESS? 

Two Copies Received 

APK 22 1907 



/ 74 oo to 

COPY B. 


Copyright, 1899, 1907, 

BY 

WALTER LORING WEBB. 



ROBERT DRUMMOND, ELECTROTYPER AND PRINTER, NEW YORK 







PEEFACE. 


■ 

Tbis book is the outgrowtli 3f the difficulties experienced by the 
author in teaching the first elements of instrumental practice to 
engineering students. The frequent practice of taking a dozen or 
twenty students into the field to do a piece of work which can 
hardly keep more than four or five men busy results generally in the 
work being really done by the few energetic ones who are anxious 
to learn, while the rest stand around and pick up what they can— 
which is frequently an insignificant quantity. 

These problems, which have had the benefit of revision after two 
years’ use, are designed to keep each student busy at some definite 
work, for which* he has definite instructions, and to utilize so thor¬ 
oughly the- limited time usually available for this work, that he will 
have a working knowledge of all the more common engineering 
instruments by the time he is ready for more advanced work. The 
student’s attention is called to the degree of precision that is required 
in instrumental work in order to obtain any required precision in 
the final results—a feature often neglected. Systematic note-keep¬ 
ing, for which forms are given, is required of the students. 

This book is not intended to replace any general text-book on Sur¬ 
veying. It aims to supplement the general instructions of such text¬ 
books by such definite directions that the students may work alone, 
aided only by sucn occasional help as may be given by one instructor 
looking after several parties, and thus solving the vexing problem of 
properly looking after the work of a large number of students with 
a limited corps of instruction. The work of many of the problems 
(e.g., chaining, levelling, angle measurements, etc.) may be so sys¬ 
tematized by means of permanent hubs on the surveying ground 
that the instructor may instantly judge of the character of the work 
done by a mere inspection of the notes submitted. 

As these instructions were written primarily for the use of the 
students of the University of Pennsylvania, they contain some local 

iii 



Iv 


allusions, but even where these occur it will be very simple to use 
the directions in a general way and apply them to any locality. 

Of course the number of these problems could be increased in¬ 
definitely, but as the time available for this work is generally some¬ 
what limited, it is deemed best to give the student only such 
problems as will most speedily and effectually teach him the capa¬ 
bilities and limitations of his instruments. Any student who works 
out satisfactorily the full list of these problems will have a better 
working knowledge of engineering instruments than many engi¬ 
neers whose instrumental experience has been confined solely to the 
commercial routine of surveying work. 


PREFACE TO FIFTH EDITION. 


Practical experience in drilling students in the use of the vari¬ 
ous kinds of engineering instruments has shown that much of the 
routine explanation which is required by each class might be 
avoided by some general instructions on the handling of instru¬ 
ments. An introduction has therefore been inserted, which, if 
thoroughly studied, will relieve the instructor from a considerable 
portion of the instruction which all students seem to require when 
first handling the instruments. 




CONTENTS 


PAGE 

General instructions in the use and characteristics of engineering instru¬ 
ments: 

Intrumental meassurements. 1 

Line measurements. 2 

Compass. 5 

Level . 9 

Transit. 12 

Routine methods of procedure for stadia work. 14 

Sextant. 15 

Polar Planirneter.. 19 

General instructions for executing different problems. 25 

Problems : 

1. Linear measurements, link-chain. 28 

2. “ “ , steel tape... 29 

3. “ “ , testing chain (or tape)... 30 

4. Chain-surveying, laying out right angles . 31 

5 . “ “ , “ “ oblique angles. 32 

6. Compass practice, sighting and reading needle. 34 

7 . “ “ , errors of sighting, setting by needle and lining in... 36 

g. “ “ , running out a line having a constant hearing.. ..... 38 

9. Level practice, reading rod and setting target.. 39 

10. “ “ , differential levelling..'. 40 

11 . “ “ , profile levelling.. 41 

• 12. «i‘ “ , testing adjustments. 42 

13. Transit practice, angle measurements. 45 

14 . “ “ .traversing .. 46 

15 . “ “ , rod alignment, setting alidade. 47 

16 . “ “ , testing adjustments . 48 

17 . “ “ , stadia readings. 50 

18 . “ “ , traversing with stadia. 51 

19. Plane-table practice. 52 

20. Pantograph. 53 

21. Polar planirneter.’. 54 

22. Barometric levelling. 56 

23. Vernier construction.*. 61 

24. Slide-rule construction. 62 




































VI 


Problems: FAGB 

25. Railroad surveying, simple curve. 64 

26 . “ “ , “ “ , vertex inaccessible. 66 

27 . “ “ , transition curve (Searles’ Railroad Spiral). 67 

28. “ “ , cross-sectioning. 68 

29 . “ “ , setting slope-stakes. 70 

30. “ “ , turnout, from existing track. 72 

31. Level trier. 74 

32. Inequality of telescope collars or of telescope axis pivots. 78 

33. Theodolite with micrometers : exercises and tests. 80 

34. Modulus of elasticity of steel tape. 84 

35. Azimuth, using solar attachment. 86 

36. Azimuth, from altitude and declination of the sun, using transit. 88 

37. Latitude, from circummeridian altitude of sun, using transit. 96 

38. Sextant practice, horizontal angles.100 

39. Sextant practice, testing adjustments. 101 

40. Latitude from circummeridian altitudes of the sun, using sextant.104 

41. Time and longitude....106 

42. Constants of a level of precision. 110 

Appendices : 

A. Probable error, formulae and use. ........ \ 14 

B. Azimuth. 116 

C. Eccentricity and errors of graduation of a graduated circle . 122 

D. Notes on the management of a students’ topographical-hydrographical 

survey. 126 

E. Notes on the management of a students’ railroad survey. 134 

Tables : 

I. Reduction of barometer reading to 32° F. . 59 

II. Barometric elevations.. . 60 

III. Coefficients for corrections for temperature and humidity. 60 

IV. Squares of numbers.. 

V. Logarithms of numbers. 146 

VI. Logarithmic trigonometric functions. 148 

VII. Natural sines and cosines.. 156 

VIII. Mean refractions. 159 

IX. Effect of refraction on declination. 160 

X. Errors in azimuth. 161 

XI. Reduction to the meridian. j62 

XII. Apparent dip of the horizon. 162 










































GENERAL INSTRUCTIONS IN THE USE 
AND CHARACTERISTICS OF ENGIN¬ 
EERING INSTRUMENTS. 


INSTRUMENTAL MEASUREMENTS. 

One of the chief objects of this text-book is to give to the student 
an adequate idea of the degree of accuracy which is necessary and 
desirable in instrumental measurements, and also the degree of effort 
which is required to attain such accuracy. The story of the young 
man who paced off the diameter of a circle and then calculated its 
circumference to six places of decimals is an illustration of the kind 
of work which is constantly being done. The length of a line is 
measured and recorded to hundredths (or even thousandths) of a 
foot, when the method used involves a probable error which is 
greater than a hundredth of a foot on each tape- or chain-length. 
Levels are read with a target to thousandths of a foot with an 
instrument so handled or so out of adjustment that there is prob¬ 
ably an error of one or two hundredths on each sight. The real 
trouble is generally caused by unbalanced accuracy. Some of the 
work is done with painstaking care and in the very next step some 
inaccuracy will be permitted which utterly wipes out the effect of 
the previous precision. By taking a large number of independent 
observations which should nominally give the same result (or which 
may be mathematically reduced to the same nominal result) and 
noting the discrepancies, an idea may be obtained of the accuracy 
of the individual results. But some ingenuity is required to insure 
that the observations are really independent. For example, if a 
line is measured several times, marking the positions of the ends of 
the successive tape-lengths on the tops of stakes, all of the inter¬ 
mediate stakes should be pulled between each set of measurements, 
or else the marks should be made on strips of paper or sheet metal, 
which should be changed between each set of measurements, so that 
the observer is uninfluenced by any previous work. When a hori¬ 
zontal angle is measured with a transit, independent readings should 
be taken on different parts of the limb so that the readings as taken 
are actually different, the value of the angle being determined by 
differences. A similar method may be used for testing level read¬ 
ings, so that the various rod readings shall be different and yet may 



be reduced to the same nominal reading. One-thousandth of a 
foot in 300 feet subtends only 0.7 second. An ordinary Wye-level 
bubble is graduated with about 20" to 30" of arc for each division. 
0.001 foot at 300 feet therefore corresponds to about 1/36 of a divi¬ 
sion of such a level bubble. An inaccuracy of half a bubble division 
therefore corresponds to nearly 0.02 foot on the rod at 300 feet. 
Even the cross-hairs of the level may be so coarse that they will 
cover 0.01 foot at a distance of 300 feet; with a focal length of 12 
inches, a cross-hair diameter of 0.0004 inch will do this. 

It must not be understood that such elaborate methods of obtain¬ 
ing a large number of independent results are advocated for practical 
commercial work. But such drill-work gives the student an oppor¬ 
tunity (which he seldom has later) of testing the character of his 
work, and teaches him what degree of effort is necessary and desir¬ 
able to reach a certain degree of accuracy, so that on the one hand 
he need not waste his time uselessly when only approximate results 
are needed, and on the other hand may use all necessary precautions 
to insure precise work when it is needed. 

The following comments are intended to supplement the instruc¬ 
tions given in the Problems. They consist chiefly of those practical 
general directions which experience has shown to be necessary when 
directing students in field-work. A careful study of these directions 
will not only save the student from many blunders, but they may 
relieve the instructor from much tiresome repetition. 

LINE MEASUREMENTS. 

Link chains. These have been almost entirely replaced by 
steel tapes, but since many are still in use, the student must be able 
to handle them properly. The chain should have been folded up 
by picking it up at the middle point (generally indicated by a round 
brass tag) and folding up four links at a time, placing the links 
crosswise to the others so that they will have an “hour-glass" 
shape, as is shown in the illustration. A strap or heavy cord fast¬ 
ened around the center will then hold them together. To use the 
chain, select a clear stretch of ground of somewhat more than half 
a chain length. Then with the bulk of the chain in the right hand 
and holding the handles with the left hand, throw the chain out. 
If the chain has been properly folded up, a little practice will suffice 
to stretch the chain (doubled) along the ground. Picking up one 
handle, the chain can then be straightened out with little or no 


3 


trouble from kinking. To guard against one source of inaccuracy 
as well as a possible injury to the chain, it should be examined 
link by link as it lies on the ground to make sure that there is no 
kink at any joint. 

Never jerk a chain. It can be moved as desired without much 
effort unless it is caught. A jerk when it is caught frequently 
means a break. Even if you can see with certainty that it is not 
caught, a jerk is useless. A perfectly steady pull, with the rear 
end held rigidly at the pin or stake just set, is an absolute essential 



Fig. i. 

to good work. The two chainmen must learn by experience how 
much tension should be used; then when the rear chainman pulls 
back with a greater tension than the normal the front chainman 
should yield, for this should mean that the chain must be adjusted 
backward to bring the rear end to the proper point; if the tension 
is less than normal, this should mean that the front chainman 
should pull it forward until the sudden increase in the tension 
shows that the rear end has reached the proper point. With a 
little practice two chainmen can quickly become accustomed to 
each other so that the two essentials of proper position and proper 
tension are quickly obtained. 

Do not stand facing in the direction of the tape and leaning back¬ 
ward, depending on the tape for support. Stand sidewise so that 
the line is not obstructed; stand squarely on both feet so that a 
perfect balance is obtained regardless of the tension on the tape; 
by leaning the right elbow on the right leg just above the knee. 




4 


any proper tension can be easily given to the tape, while the align¬ 
ment may be readily seen. One of the chief advantages of the 
attitude is its ease and steadiness. If the end of the tape must be 
held up high so as to be on a level with the other end, of course a 
plumb-bob must be used. Wind the plumb-bob string around the 
tape precisely on the zero mark—or over the end of the ring if that 
is the zero mark. Then the eyes may be steadily fixed on the 



plumb-bob, either to note its position or to adjust it to the position 
of the previously established point. 

Steel tapes. Much of the above instruction applies equally 
to the use of steel tapes. A tape is much more liable to be broken 
than a link chain, but the soft-steel tapes recently manufactured 
are capable of very rough usage. During wet weather or when a 
tape has become wet and muddy, it should be carefully wiped with 
an oily cloth before it is rolled up on the reel. Neglect to do this 
will quickly cause it to become rusty. 

The student should learn to exercise special care that the tape is 
always held as truly horizontal as possible. Great care is often 
used to line in the tape with a transit, even using tacks on the tops 
of stakes to make sure that no error will arise from that cause, but 
at the same time the two ends of the tape have a difference of level 
of perhaps a foot or more. Precision in this case is simply wasted. 
A difference of level of one foot, which would frequently be ignored, 
will cause an error one hundred times as great as an error of one- 
tenth of a foot in horizontal alignment, an error which usually 
would be carefully avoided by transit alignment. The difficulty 
in holding the two ends at the same level is largely due to one’s 





5 


inability to correctly estimate the rate of slope, an optical illusion 
causing a truly level line on a hillside to appear sloping. Another 
inaccuracy is due to the desire to use full chain lengths of 100 feet 
rather than “ break chain.” A 6% or 7% grade does not appear very 
steep and so full 100-foot lengths will be attempted, but the imprac¬ 
ticability of holding the down-liill end 7 feet above the ground, and 
the difficulty of holding it even 6 feet above the ground, generally 
causes an error of a foot or more in level, especially as the chainmen 
will honestly think that the tape is really level. The amount of 


Height of Triangle. 

Base. 

Hypothenuse. 

Error. 

10 feet 

100 feet 

100.499 feet 

0.499 foot 

5 “ 

100 “ 

100.12499 “ 

0.12499 “ 

1 “ 

100 “ 

100.00500 “ 

0.00500 “ 

0.5 “ 

100 “ 

100.00125 “ 

0.00125 “ 

0.1 “ 

100 “ 

100.00005 “ 

0.00005 “ 


error due to a difference of level of the two ends is shown in the 
tabular form. With the extreme difference of 10 feet the error is 
about 6 inches. The error varies about as the square of the differ¬ 
ence of elevation. With one foot difference the error is 1/16 inch. 
Even with 0.1 foot difference of elevation (or of alignment) the 
error is less than six ten-thousandths of an inch. This shows the 
utter futility of great refinements in horizontal alignment under 
the ordinary methods of measurement. 


COMPASS. 

Uncertainties of compass readings. One of the principal 
objects of Problems 6, 7, and 8 is to demonstrate practically to the 
student the uncertainties and inaccuracies of compass work. For 
some kinds of surveying work where simplicity and rapidity are 
prime essentials and in which a considerable degree of inaccuracy, 
or at least uncertainty, might be tolerated, the compass is an exceed¬ 
ingly useful instrument. As an adjunct to transit w T ork, to check 
the angle work, and to detect gross errors or the slipping of the 
plates, it is also useful, and its use in this way should not be neg¬ 
lected. But the student should be fully aware of the limitations 
of its use. The presence of large masses of iron and steel in build- 














6 


ings, as well as high-tension electric currents, limits its use in cities, 
and even barbed-wire fences and trolley lines along country high¬ 
ways introduce a local attraction when attempting to use a compass 
on such boundary lines. Even assuming that the diurnal, the 
lunar, and the annual variations, as well as the normal declination, 
have been duly allowed for, there still remain local attractions of 
an unknown and uncomputable amount, which are often so great in 
magnitude that they utterly overshadow all the others. These may 
be due to high-tension electric currents, masses of iron or iron ore 
(known or unknown), or an “electric storm.” The only certain 
way of avoiding such errors is to take foresights and backsights on 
all lines. The closeness of the agreement of such lines will be a 
very fair test of the presence of local attraction. 

Setting up the compass. Since there is far less necessity for 
precision in the leveling of the compass plate, a compass is usually 
provided with a ball-and-socket joint. Although this is more 
troublesome to an inexperienced man than leveling-screws, it is 
possible to level up much more rapidly. The screw which clamps 
the ball should be screwed up until the friction is sufficient to hold 
the plate in any position and yet permit it to be move£ and adjusted 
without great force. Only practice will show just how much this 
should be. After the plate is leveled the screw may be tightened 
still more, but it must be done with great care or the plate will be 
thrown out of level while doing it. Perhaps the better plan is to 
have the screw so tight, even during the process of adjustment, 
that it will not slip while the instrument is being turned about its 
axis. 

A compass tripod is not usually provided with a plumb-bob or 
with a centered hook for a plumb-bob string. A compass ring is 
usually divided to half-degrees, and an estimation to sixths of a 
division (or 5' of arc) is about as close as the needle can be read. 
Five minutes of arc at a distance of 500 feet means nine inches. 
Even if the compass is set up over a stake, it can easily be centered 
by eye to within an inch or so, which is far within the accuracy of 
the needle reading. When, as is.so common, an offsetted line is 
run parallel to a fence, the offset can be measured directly from the 
instrument. A plumb-bob is therefore a useless refinement. 

Setting up a tripod. Although it is possible to level up an 
instrument, and especially a compass which has a ball-and-socket 


7 


joint, when the plate of the tripod head is considerably out of level, 
yet it is generally possible, even on very rough and steep ground, 
to so plant the three legs that the tripod plate will be practically 
level and very little adjustment will be necessary. The ability to 
quickly select positions for the tripod legs which will make the 



tripod plate practically level is largely a matter of experience, but 
there are certain principles which the student will do well to keep 
in mind and which will enable him to more readily acquire the 
skill to set up .a tripod on uneven ground. The following discus¬ 
sion presupposes the use of the ordinary plain tripod with fixed 
legs. Extension-leg tripods look very attractive, but their positive 
disadvantages should not be forgotten. (1) There is always the 
danger that the clamps may not have been clamped so tightly but 
that they may slip. (2) Time may be wasted in adjusting the clamps 
when a little skill and ingenuity would set a plain tripod in place 
in far less time and with no danger of slipping. (3) Except in 
mines and tunnels, the occasions are rare when a plain tripod can¬ 
not be set up so that the tripod plate is practically horizontal. On 
a steep side hill, where the slope is approximately uniform, the best 
general method is to place two legs on the same contour, which is 
a little below the stake and equidistant from the stake. The third 
leg will be on the up-hill side on a line running almost directly up 
from the stake. If the side hill were actually plane, there is but 
one position for the third leg. Slight inequalities of the surface 



8 


can then be easily allowed for by mere adjustment. When the 
side slope is very steep, it may even be necessary to point the third 
leg with an upward slope rather than a downward slope. If the 
two lower legs have been firmly planted, this position will be found 
to be sufficiently rigid. The student should learn by trial and 



Fig. 4. 


experience that, when the tripod plate is rery far from being level, 
it is often true that moving one leg-point in an arc of a circle about 
the stake, and without disturbing the centering of the instrument 
over the stake, will suffice to make the plate level. The student 
should test this by noting that when a tripod has been properly set, 
with the head level, the shifting of the point of one leg will throw 
the head far out of level and that it may easily be made level by 
the proper adjustment of that leg. 

The above remarks apply to the setting up of all tripods. In 
the case of the level tripod it is all that is necessary, except when 
the elevation of the telescope is of importance. But with the 
compass tripod, and still more so with the transit tripod, the center 
of the tripod head must be vertically over the stake or hub. The 
student must train himself against the optical illusion on a side 
hill, which makes him think that the tripod head is directly over 
the stake when it is really a little too far down the hill. 

When the tripod legs are provided with wing nuts where the 
legs are fastened to the head, as the best tripods are now made, 
the nuts should be tightened whenever the instrument is set up, 
and loosened again just before it is picked up for .removal. The 




9 


legs cannot be turned on the head unless the screw is somewhat 
loose and then the tripod is far less steady than when the wing 
nuts have been screwed up tightly. This is the feature that makes 
tripods having wing nuts better than the old-fashioned kind. 

Compass readings. Text-books are apt to emphasize, per¬ 
haps somewhat unduly, the magnetic attraction of small articles 
of steel which one is apt to carry with him, as, for example, keys, 
knives, key-chains, the steel band which may be in the rim of a 
derby hat, axes or hatchets which may be lying around, the steel 
shoes of aligning-rods, etc. While it is always better to be on the 
safe side and to make sure that the needle is not affected by any of 
these things, it is quite possible to waste time and effort in removing 
such objects from the neighborhood of the instrument when it is 
wholly unnecessary. It should be remembered that the intensity 
of such attraction (or repulsion) varies as the square of the distance 
of the object from the needle. The steel in a hat-brim would be 
very likely to influence the needle, since the brim of the hat might 
approach very, near to the compass-box while the needle was being 
read, while a bunch of keys whose magnetism and mass might be 
far greater would have no influence, since it would be two or three 
feet away. The student will do well to test the effect of any object 
by taking.careful readings of the needle when the suspected object 
is placed alternately on diametrically opposite sides of the instru¬ 
ment. The difference of readings should be about twice the influ¬ 
ence of the attraction. If no appreciable difference is observed, it 
shows that that particular object has no appreciable effect when it 
is at that distance from the instrument. 

Care should be taken that the eye which reads the needle is 
located in a vertical plane through the needle, and if possible very 
nearly above the needle-point in order to avoid “parallax.” Even 
though the needle is only read by estimation to 5' of arc, or 1/6 of a 
30' space, it is much easier to estimate the fraction of a space by 
observing the location of the point with a magnifying-glass. 


LEVEL. 

Use of level. The refraction of the light through the glass 
of the level-bubble is apt to give deceptive results unless some care 
is taken to avoid parallax. The surest way to avoid parallax is to 


10 


have the eye directly over the bubble, so that the ray of light from 
the bubble to the eye passes through the glass perpendicularly. 
But it is frequently impracticable to set up the level so that the eye 
can be placed over the bubble and the bubble must be viewed from 
the side. If care is taken that both ends of the bubble are read 
with the eye in a similar position regarding the bubble, the refrac¬ 
tion of the light coming through the glass will be the same in both 
cases and its effect will be neutralized; but if the eye is so placed 
that it looks at one end of the bubble obliquely and at the other 
end perpendicularly, the resulting error may be considerable. In 
the introduction of this work comment was made on the effect on 
the final result of certain small errors in the adjustment of the 
instrument or of inaccuracies in its use. The student should learn 
by actual test, as described in Problem 12, what is the sensitiveness 
of his instrument. By this means he will know how closely the 
bubble should be adjusted to the precise center in order to obtain 
a given accuracy in the final result. He should also learn to con¬ 
stantly watch the bubble so that the effect of a microscopic settle¬ 
ment of the instrument or of its distortion by heating from the 
sun’s rays shall not alter the line of collimation to an appreciable 
extent. 

The student should learn to test the focusing of the cross-wires. 
The simplest and surest way is to sight the telescope at the sky or 
at any bright but neutral surface with the object-glass purposely 
put out of focus so that the eye is not distracted by the appearance 
of any object sighted at. Then move the eye-piece until the cross¬ 
wires are as sharply defined as it is possible to make them. The 
student should remember that there is one definite position for the 
eye-piece for his own eye regardless of any change of focusing of 
the object-glass. The eye-piece must be- so focused that the eye 
will obtain a sharply defined view of the cross-wires. Then, when 
the object-glass has been so focused that the image is formed 
exactly in the plane of the cross-wires, the focusing is complete. 
The student should not permit himself to be confused by the fact 
that he can so focus the object-glass and eye-piece that he obtains 
a distinct view of the object sighted at even when the cross-wires are 
out of focus. This happens when the image formed by the object- 
glass does not lie in the plane of the cross-wires. Even though the 
cross-wires appear fairly distinct, they may be sufficiently out of 


11 


focus to cause an error in the instrumental work. This may be 
tested by setting the telescope so that the cross-wires cover a very 
well-defined point. Then move the eye up and down or crosswise 
and note whether the cross-wires apparently move over the object. 
If they do, the cross-wires are not in focus. Therefore the only 
safe rule is to focus the cross-wires as previously described. After 
the eye-piece has once been properly focused for any one eye it 
need not be changed no matter how much focusing may be required 
of the object-glass to adapt it to distant or near objects. 

As a part of the general problem of studying the characteristics 
of his instrument the student should obtain a practical idea of the 
coarseness of the cross-wires, which practically means, in this case, 
how much space is covered on the rod by the cross-wire when the 
rod is at a distance of two or three hundred feet. When the cross¬ 
wire is very coarse, as is sometimes found with a cheaper grade of 
instrument, the space covered on the rod is very considerable and 
it makes the work correspondingly uncertain. 

Use of target-rod or “self-reading” rod. Many a student 
and many even of those who call themselves engineers waste con¬ 
siderable time in using a target when a self-reading rod would be 
sufficiently accurate for the purpose and also much more expeditious. 
The student should remember that much of the best geodetic level¬ 
ing is done with self-reading rods, and it is even open to argument 
whether a self-reading rod is not the best to use under all circum¬ 
stances. Of course, a target is almost an essential with a “New 
York ” rod or “Boston” rod, since the marks on the plain rod are 
not visible for any considerable distance, but with a “ Philadelphia ” 
rod or any other form of painted rod the graduations are plainly seen 
for as great a distance as it is practicable to do accurate work, and 
but little skill is required to estimate the readings to the nearest 
hundredth of a foot. This has as great a degree of accuracy as is 
of any practical value for the great bulk of ordinary leveling work. 
In the introduction of this article there are given some figures on 
the accuracy necessary in order that target readings to the thousandth 
of a foot may have any practical value. The student should there¬ 
fore rely on using a self-reading rod for the great bulk of all level¬ 
ing work. When it becomes necessary to do the leveling with 
such accuracy that target readings to a thousandth of a foot are 
used, the student should take corresponding care that his instru- 


12 


ment is in perfect adjustment, that the bubble is always precisely 
in the center when the reading is taken, that allowance is made, if 
necessary, for the thickness of the cross-hairs, and that every step 
of the work is taken with such careful accuracy that the readings 
to thousandths are justifiable. Unless such care is taken, it merely 
gives a false appearance of accuracy to the work to use a target 
and to read to thousandths of a foot. 

TRANSIT. 

Setting up. The subject of setting up a tripod, especially 
on sloping ground, so that the tripod plate shall be approximately 
horizontal, has already been discussed. Many a good instrument 
has had its leveling mechanism nearly spoiled by bad practice in 
leveling up. The large majority of transits and levels are provided 
with four leveling-screws, which oppose each other in pairs. It is 
quite possible for one who has strong fingers to turn a leveling- 
screw so hard that the leveling-plate becomes actually bent. 
When the screws of one pair have been turned against each other 
so that they hold the plate very tightly in the ball-and-socket joint, 
it becomes difficult to move the plate in the other direction by 
means of the other pair of leveling-screws. If at any time it be¬ 
comes necessary to change the leveling of the instrument very 
greatly, it is always better to loosen all four screws until the instru¬ 
ment turns very readily in its ball-and-socket joint; then, by easy 
motion of the screws of each pair, the instrument can be quickly 
and easily leveled until the leveling is nearly accurate. The final 
step will be to screw the leveling-screws firmly against the tripod 
plate so that the instrument is properly leveled and yet there is no 
undue stress in the leveling-plate. 

Horizontal-angle work. In cases where the horizontal 
angles are not required with minute accuracy, it may be justifiable 
to use only a single vernier (especially if the instrument is so well 
made that its eccentricity never exceeds one-half minute of arc) 
and to set the vernier at 0° as an initial reading; but when the 
greatest accuracy is required, not only should both verniers be read, 
but it is better to sight the instrument with the verniers reading 
any reading. The verniers on the horizontal plate of a well-made 
sharply graduated transit may be easily read or estimated by means 


13 


of a microscope to the nearest half-minute of arc. By reading both 
verniers and taking their meau, in case the opposite readings 
should differ by even one half-minute, we have a mean value to 
the nearest 15 seconds. 

A 6-inch circle has a circumference of 18.85 inches, but 15 seconds 
of arc is of the circumference, which means but .00022 of an 
inch in this case. In order to set the vernier at precisely 0° to 
within 15 seconds of arc, the slow-motion screw must be turned so 
precisely that its position is located to within twenty-two hundred- 
thousandths of an inch. Although the tangent screw may be turned 
so that no variation from 0° is observable even with the microscope, 
there is always the chance for that psychological error of believing 
that the reading is some specially desired reading if the inaccuracy 
is really small. When the verniers are set at any reading, the 
mind is not biased by thinking that the reading must be any partic¬ 
ular reading, but, in an unprejudiced way, can make the reading 
of each vernier to the nearest half-minute with the least probability 
of error. 

The educational value of Problems 13 and 15 are very great 
because they give to the student a practical idea of the accuracy 
obtainable by a certain degree of effort with the instrument used. 
A test that is even more simple and which does not require the 
service of a rodman may*be made by setting up the transit, clamp¬ 
ing the lower plate as rigidly as possible, and sighting at some well- 
defined point located at a great distance. After reading the vernier, 
the upper plate is loosened and the alidade is swung around slightly. 
Again set the vernier at the same reading and observe how closely 
the vertical cross-wire again sights at the distant point. Again 
loosen it and repeat the operation several times. Of course, it is 
possible that some of the resultant discrepancies will be due to an 
actual disturbance of the instrument as a whole, but unless unusual 
skill and care are shown in the setting of the instrument at the 
constant vernier reading there will be discrepancies which will 
frequently be a revelation to one who has not made such a test. 

Vertical-angle work. When the transit is to be used only 
for measuring horizontal angles, any reasonable care.in leveling the 
horizontal plate will suffice to avoid any measurable error owing 
to lack of horizontality of the horizontal plate, but many instru¬ 
ment-men do not realize the care that must be taken to have the 


14 


instrument truly level when a vertical angle is to be measured. 
The plate bubbles are always somewhat coarse and it is almost 
impossible to adjust them so that there is any feeling of certainty 
that the plate is level even to one minute of arc. No vertical-angle 
work should be done with a transit unless it is provided with a 
“long bubble” under the telescope, and even then it is advisable 
to check the adjustment, at least every day that it is used, as 
follows: After leveling the instrument as carefully as possible by 
means of the plate bubbles, clamp the telescope so that it is horizontal 
and with the slow-motion screw bring the long bubble exactly to 
the center. Then swing the whole instrument 180° about its vertical 
axis. The bubble should again be in the center, but if it is not, it 
means that the vertical axis of the instrument is not truly vertical. 
Correct half of the error by means of the slow-motion tangent- 
screw to the telescope and the other half by means of the leveling- 
screws. Theoretically this should accomplish the desired result, 
but practically it may be found necessary to make repeated adjust¬ 
ments of the tangent-screw and the leveling-screws so that the 
bubble will remain in the center when the telescope is pointing in 
any direction; then note the reading of the vernier of the vertical 
arc. If it is adjustable, it should be adjusted to 0°. If it is not 
adjustable, the “index error” should be read, paying particular 
attention to its algebraic sign. For example, if it already reads 
-f 0° 5', thus indicating that, according to the vernier, the telescope 
is pointing upward when we know it is horizontal, it would mean 
that every upward reading will be 5 minutes more than its true 
reading, and hence the index error is — 0° 5', which means that we 
should subtract 0° 5'. from every upward or positive angle and add 
0° 5' to every downward or negative angle to get the true reading. 

ROUTINE METHODS OF PROCEDURE FOR STADIA WORK. 

1. After setting up the instrument, observe immediately the 
“height of instrument,” which is the elevation of the horizontal 
axis of the telescope above the ground (or stake) immediately 
under the plumb-bob. 

2. If a target is used on the stadia-rod, have the rodman set 
the target at the reading of the “height of instrument ” 

3. Having set the horizontal plate of the transit at the proper 


15 


azimuth, sight the vertical wire at the middle of the rod and the 
middle horizontal wire as near the target as possible and yet 
allow the lower cross-wire to be on an even foot-mark. 

4. Read the wire interval. If the lower wire is on an even foot¬ 
mark, it is only necessary to count the number of whole feet and 
the odd interval at the top. 

5. Set the middle wire on the target and dismiss the rodman. 

6. Read the vertical angle and the azimuth. 

Reducing the observation. 7. Add f+c (usually 1 foot) to 
the distance ( R ) obtained from the rod interval. For level or nearly 
level lines, this is the true horizontal distance. When a correction 
is necessary, it is found from the equation, Correction = R sin 2 a, 
which may be computed from a “stadia table” or by means of a 
“stadia slide-rule.” 

8. The difference of elevation may be obtained for all ordinary 
anglesjfrom'the equation, Difference of elevation = (R-\-fc)\ sin 2a, 
which may be computed from a stadia table or by the use of a 
stadia side-rule. 

SEXTANT. 

The sextant is in one respect the most remarkable instrument 
of precision ever invented—considering that with it angles may 
be measured to the nearest 10 seconds of arc, although the in¬ 
strument is held with the hand by an operator who is perhaps on 
a vessel or even in a small unsteady boat where the use of a transit 
would be impracticable for even the roughest work. The sextant 
is used either for measuring the “altitude” of the sun or a star, 
or to measure the angle between two points. In the latter case 
the angle measured lies in the plane determined by the two points 
nnd the instrument, and does not necessarily equal the horizontal 
projection of that angle, as would be the case if the angle were 
measured with a transit.* The sextant is the only instrument 
which will give directly the measurement of an angle lying in an 
oblique plane. 

General principles of construction. The ordinary sextant 
consists essentially of an arc of about 70° or 80°, having an arm 
(VI, Fig. 5) revolving about the centre (/) of the arc, having a mirror 
called the “index” mirror (/) attached to the arm, having a mirror 
called the “horizon” mirror ( H ) which is attached to the frame 
of the arc, and having also an eyepiece (E) which may be either 


16 


a telescope or a small tube which merely directs the line of sight. 
The horizon mirror consists partly of clear glass, so that an object 
at 0 2 may be seen through the glass and the eyepiece E. The 
sextant should be held in the right hand, the movable arm being 
adjusted by the left hand. The movable arm is provided wi h 
a clamp and slow-motion screw. 

To obtain the angle between lines from the two points 0 2 and 0 lt 
the eyepiece E must be pointed at 0 2 and the arm VI moved until 



a ray of light from O u falling on the index mirror (/), will be re¬ 
flected to the silvered portion of the horizon mirror ( H ) and from 
there reflected back through the eyepiece. The required angle 
is the angle 0 X C0 2 = A =2(3—2a. But the angle NMH=(3—a = \A. 
But since NM and MH are respectively perpendicular to IS and 
HS and also to IV and 10, the angles are'equal and the angle OIV 
=(3—a = %A. Since each angle measured is really twice the angle 
OIV on the arc, it is more simple to double the graduations on the 
arc, so that the apparent reading of the arc gives directly the true 
measure of the angle A. 









17 


The point 0 is a point on the arc such that 01 is parallel to the 
horizon mirror ( H ) or perpendicular to the normal HM. When 
IV coincides with 10, the two mirrors are parallel. The point 0 
is not necessarily the zero-point of the vernier, but the vernier 
should read 0° when IV coincides with 10. 

One theoretical defect of the sextant is the fact that the vertex 
of the angle (C) is a variable point which is somewhere within th6 
instrument for all large angles, but which will be a long distance 
behind the operator for very small angles. If the uncertainty of 
either point sighted at is as great as the perpendicular distance 
from I to HE (two or three inches), as it is in all astronomical 
work and in terrestrial work where one point is at a distance 
of a mile or more, the error involved is too small for measure¬ 
ment. 

Measurement of altitudes. As a natural horizon is seldom 
obtainable, except at sea, the use of an “artificial horizon” be¬ 
comes necessary. This is a basin containing a reflective liquid. 
Mercury is frequently used, but molasses is less disturbed by wind. 
As the free liquid assumes a level surface, a ray of light (S lt Fig. 6), 


SX 


\ 


\ 

l a \ 

_ t _i 






i- a - 

\ a 

% 


z 




_ 


Fig. 6. 


striking the liquid at an angle a, will be reflected with an equal 
angle. If the sextant, held at E, is pointed downward so as to 
catch the reflection from the liquid, and the index mirror is turned 
so as to catch the parallel ray S 2 , the angle measured will be 2a 
or the “double altitude.” 









18 


Altitude of sun or moon. When pointing at the sun or moon, 
it is practically difficult to accurately superpose one image directly 
over the other, and it is much more accurate to move the arm 
until the discs are tangent to each other. The reflection (aSJ from 
the liquid is reflected once and is therefore inverted. The other 
image ( S 2 ) is reflected twice, is therefore re-inverted, and is seen 
erect. When the image from the liquid appears just underneath 
the other image, the two images of the lower limb are just in coin¬ 
cidence, and the altitude measured is therefore that of the lower 
limb, which is less than the altitude of the centre by the angular 
value of the semi-diameter. 

Dip of the horizon. When observations are taken at sea, 
using the natural horizon, the instrument is always at some dis¬ 
tance above the water. A line to the real horizon (EH, Fig. 7) 



would therefore dip below a horizontal line (EB) by an angle a 
which is equal to the angle ECH. But the refraction of the atmos¬ 
phere complicates the solution by apparently raising up the horizon 
and also by rendering visible more of the sea than would be visible 
if there were no refraction, so that the apparent dip is less than a. 
Knowing the elevation of the sextant above the water, the apparent 
dip may be found by reference to Table XII, which gives the dip 
for all elevations from 6 feet to 30 feet. 



19 


POLAR PLANIMETER. 

The Polar Planimeter is no longer considered as being merely 
an ingenious mathematical curiosity. The instrument is capable 
of such marvellous precision, and the practical uses to which it can 
be put are so varied, that it must now form a part of every com¬ 
plete engineering equipment. The essential work of the instru¬ 
ment is to fin d the area of any plane figure, no matter how irregular 
its perimeter may be. This permits the solution of many related 
problems. The planimeter may be used as follows: 

1. To obtain the area of plotted figures which are bounded by 
irregular lines, such as drainage areas, ponds, property bounded 
by water lines, etc. 

2. To obtain the area of plotted earthwork sections, especially 
when very irregular; also profiles, indicator diagrams, etc. 

3. To obtain the average of observations taken at either regular 
or irregular intervals. This is done by plotting the observations 
as vertical ordinates at horizontal intervals proportionate to the 
intervals of the observations. The tops of the ordinates are then 
joined by an irregular line. The area of the figure divided by 
the length of the base gives the average ordinate. 

Theory. 

The mathematical demonstration of this problem is best made 
by the use of the integral calculus; but, for the benefit of those 
unfamiliar with the calculus, the following demonstration has been 
g[ ven — a demonstration involving nothing higher than elementary 
algebra, geometry, and trigonometry, although the proof is rigidly 
accurate. 

1. When the planimeter is in the position PHWC (see Fig. 8), 
the plane of the wheel, which is perpendicular to the axis PH, 
passes through C. If the instrument is revolved about C, with 
thb angle WHC ( = a 0 ) always constant, the motion of the wheel 
over the paper will have no component in the direction of its 
plane, and the wheel will not revolve. The pointer in this position 
describes the “zero circle.” 

2. When the planimeter is in the position P'H'W'C and is re¬ 
volved about C, with the angle W'H'C (=«i) always constant, 
the wheel will have a combined sliding and rolling motion. For 


20 


an infinitesimal movement (W'b) the wheel will roll an amount 
W'a and slide perpendicular to its plane an amount ab. When 
rolling in this direction, the movement is called negative. 


\ 



3. When the point P is moved from P to P', the wheel W will 
both slide and roll, but its rolling will all be in a negative direction. 

4. If the pointer were to move back from P f to P, the wheel 
would again slide and roll in precisely the same amounts but in 
contrary directions, and when it reached P it would have identi¬ 
cally the same position, and the reading of the index would be 
identical with the previous reading at P. 

5. If the pointer were to move from e to d, the amount and 
direction of both the slipping and rolling would be the same as when 
it moved from P' to P. 

6. If the pointer were to start from P, move to P', thence to e, 
thence to d, and thence back to P, the resiiltant$rolling of the wheel 
is the same as that for the line P'e alone; for the rolling for dP is 
zero (§ 1), and the rolling for PP' will be just neutralized by that 
for ed (§§ 3, 4, and 5). 




21 


7. Therefore when the pointer is moved to the right on the arc 
of a circle within the zero circle, about C as a centre, the indica¬ 
tion is negative and is the same as if the pointer moved around 
the area included between the arc, the corresponding arc of the 
zero circle, and the including radii. 

8. If the pointer moved in the opposite direction, the indication 
would be the same in amount but 'positive. 

9. By similar demonstrations, similar facts may be shown for 
any other elementary area, except that - 

(a) When the pointer is outside the zero circle and moving to the 
right, the indication is positive. 

( b ) When outside and moving to the left, the indication is nega¬ 
tive. 

10. The perimeter of any area may be considered as made up 
of a combination of infinitesimal arcs and radial lines having the 



fixed point of the planimeter as centre. Its total area is the 
algebraic sum of all the infinitesimal areas lying between each 
arc and the zero circle. 

11. If the pointer of the planimeter moves around each infini¬ 
tesimal area in turn in such a manner that when moving on the 
perimeter it moves in the same direction as though moving con- 



22 


tinuously around the perimeter only, the pointer will move over 
all interior lines an even number of times in opposite directions* 
Therefore the accumulated registration of the wheel will be the 
same as though it moved on the perimeter only, for all registration 
on interior lines will be neutralized by the equal motion on them 
in opposite directions (§§ 4 and 6). 

12. Referring to Fig. 8, P'e = CP'Xp='/m 2 +l 2 +2ml cos a.X/?. 
W'b = CW' X /?. The rolling of the wheel = TF'a (§ 2). W'a = 
(W'hX W'b) + CW' = (n—m cos cq)^, since W'b+CW'=p and W'h 
=n—m cos cq. 

Area PP'ed = \(PCxp)PC-%(P'CXp)P'C 

= £/9(PC 2 -P'C 2 ) 

= §/?[(Z 2 +n 2 +2nZ)+ (m 2 —n 2 ) — (m 2 +l 2 -\-2mlcos aj] 
={3l(n—m cos a) 

= lXW'a (i.e., I times the rolling of the wheel). 

13. When the pointer moves around an elementary area bounded 
by an arc, the corresponding arc of the zero circle, and by the 
two bounding radial lines (all having C as centre), the resultant 
motion of the wheel is the same as though it moved on the arc 
alone (§ 6); the wheel rolls a distance equal to the area divided 
by Z (§ 12). If it moved in turn around each elementary area of 
a large area, the resultant motion of the wheel would be the same 
as though it moved continuously around the perimeter of the 
large area (§§ 10 and 11), and therefore the total resultant motion 
of the wheel will equal the area of the figure divided by Z. 

14. Therefore if c=the circumference of the wheel, n the number 
of turns recorded by the index, and Z the length of the arm from 
F to P, then 

Area = Inc. 

15. Fixed centre inside the figure. If the pointer is moved 
around the perimeter A (Fig. 10) to the right , the indication 
will be positive (§ 9), but will indicate only the area between A 
and the zero circle. Therefore the total area will equal the in¬ 
dicated area (Inc) plus the area of the zero circle \n(m 2 -\-1 2 + 2nl)\ 
If the pointer is moved to the right around perimeter B, the record 
will be negative (§ 9) and will correspond to the area between 
B and the zero circle. Therefore the algebraic sum (the numerical 
difference) will give the true area. 



23 


16. To find the area of the zero circle. The accurate measure¬ 
ment of Z, m, and n, directly from the instrument, is imprac¬ 



ticable, but the area of the zero circle may be obtained from the 
consideration that it is equal to the algebraic difference (the 
numerical sum) of the readings when an area is measured (1) 
with the fixed centre outside the figure and (2) with the fixed 
centre inside the figure. Therefore draw some figure large enough 
so that the fixed centre may be placed inside and the pointer 
may travel all around the perimeter, and yet not so large but that 
the fixed centre may be placed outside and the pointer may reach 
all parts of the perimeter. If the pointer is always moved around 
the figure to the right, the reading when the fixed centre is inside 
will be negative, and when the fixed centre is outside will be positive. 
Their algebraic difference, which is their numerical sum, will be 
the area of the zero circle. A 6-inch square will be the best figure 
for this purpose, for reasons given below. 

Practical Use. 

These instruments are generally constructed so that the arm 
PH (the length Z) may be made variable, and the arm is graduated 
so that, by setting it at given marks, the wheel will give the area 




24 


directly in almost any desired unit. If the desired unit is not 
marked on the arm, draw a square having 1, 2, or 3 of the desired 
linear units on a side and take its reading. The reading for this 
square will be 1, 4, or 9 times the reading for one of the desired 
square units. Then the reading for the given irregular figure 
divided by the reading for one square unit will give the number 
of such square units. 

When running the pointer along straight lines it will increase 
the accuracy to guide the pointer by running it along a straight 
edge. Therefore in obtaining the area of trial unit figures, or 
that of the zero circle (§ 16), it is best to use straight-lined figures. 

If there is any doubt of the accuracy of the markings on the 
arm PH, they may be tested by using a test-bar that generally 
accompanies the instruments. This bar has a needle centre and 
drilled points at even inches or centimetres from the needle. By 
placing the pointer in one of these drilled points, an accurate circle 
may be described by the pointer about the fixed needle centre. 
This circle has a known area, and the wheel reading should corre¬ 
spond. 

The previous work has assumed that the initial reading is zero, 
and then the final reading is the true “reading ” for the area. 
The practical difficulty in having the initial reading 'precisely zero, 
when the pointer is at the desired starting-point, justifies the 
practice of taking both initial and final readings (whatever they 
may be) and then taking the difference as the “reading ” or measure 
of the area. 

Care must be taken, especially in work involving the zero circle, 
to note the direction of motion of the index wheel and also whether 
it runs past the zero of that wheel and how often. When the 
fixed centre is inside the figure, the index may turn completely 
around tw T d or three times—turning negatively or backward. 
Then if the initial reading is (say) 4.642, call it 34.642. Then 
watch the wheel run back through zero, and it is in the twenties; 
again through zero, and it is in th e “teens”; again through zero, 
and it is in the single units, and the reading is (say) 8.796, 
Then (34.642—8.796) is the true difference or negative reading. 

Problem 21 gives some practical exercises illustrating these 
principles. 


GENERAL INSTRUCTIONS FOR EXE¬ 
CUTING DIFFERENT PROBLEMS. 


1. Complete notes of each problem, in proper form, tabulated 
whenever possible, must be made during the progress of the field¬ 
work. 

2. The notes should be so complete that the significance of every 
recorded number should be easily intelligible to any one conversant 
with the subject. 

3. All calculations must be completed when reports are submitted. 

4. Note-books must be turned in each day with the instruments— 
except when the problem requires extended computations on the 
data obtained in the field, in which case two days’ extra time will be 
allowed. 

5. The note-book should be a journal of each day’s field-work— 
giving (a) date, ( b) time of beginning and ending work, (c) number 
of problem, ( d) personnel of party, and then a detailed report of 
individual work. 

6. No notes are to be copied from another’s note-book, except such 
figures as target readings of a level rod, etc., taken by the assistant, 
which will be necessary for a complete report on the problem. With 
such exceptions each student reports only his own individual work. 

7. When a problem requires the work of two or more (as is usually 
the case) each student in turn must act as chief, do the instrumental 
work, and take notes for a complete solution of the problem. 

8. Even when a student acts only as assistant during any one day’s 
work, a report must be submitted by him giving date, time of work, 
number of problem, and personnel of party. 

9. No credit will be given for field-work except as it can be esti¬ 
mated from the notes. No verbal reports will be credited. 

10. Read the instructions carefully before beginning the field-work 
on any*problem; be sure that all necessary equipment is provided 
and that the instructions are understood. 

11. Students are held responsible for all equipment issued to 
them. Therefore, examine all equipment as it is issued and report 
immediately any injury or deficiency that may be discovered, in 
order that the responsibility may be properly located. 


25 














































* 


























* 










l 


w 




% 







{ 






♦ 












I 














































* 




















































PROBLEMS IN THE USE AND ADJUSTMENT 


OP 


ENGINEERING INSTRUMENTS. 


28 


- 1 — 

LINEAR MEASUREMENTS—LINK-CHAIN. 

Equipment: Liuk-cliain, set of marking-pins, two transit-poles, 
two iron plumb-bobs. 

Location: When possible, these measurements should be taken 
between points (or permanent hubs) which are about 1000 feet apart 
and where the intervening ground has asleep and variable slope. 

Method : Measure the distance three times, lining in by eye, taking 
precautions to have the two ends of the chain always at the same 
level, “ breaking chain ” if necessary, etc. 

Always drag the chain ahead its full length, even if it is necessary 
for the head chaimnau to walk back to “ break chain.” Always 
place a pin at each even hundred feet, and begin again at that point 
for a new chain length. Care should be taken that no greater 
length of chain should be used than can be held truly level without 
holding the down-hill end inconveniently'high. The student should 
learn that an optical illusion will cause a truly level line to appear as 
if it sloped down (opposite to the slope of the hill), and that when 
a chain is stretched down a hillside and held up so that it appears 
to an inexperienced eye to be level it is probably sloping down the 
hill. The pins may be aligned by eye, aided by the plumb bobs. 

Each student makes a set of three measurements acting as head 
chaimnau. The extreme range of the three values should not be 
more than of the distance. 

Form of Notes for Problems 1 and 2. 

[left-hand page of note-book.] 


Problem 1.* 


Line. 

Meas. 

Slope. 




a 15-A 22 
22- 15 
15- 22 

1087.4 

1086.9 

1087.6 

down-hill 
up- “ 

down- “ 





♦The number of problem, date, time of work, and personnel of party, which 
are here illustrated, should be recorded in the Notes for all problems, but they 
will not hereafter be illustrated in the Forms of Notes. 















29 


- 2 — 

LINEAR MEASUREMENTS—STEEL TAPE. 

Equipment: 100-foot steel tape, two transit-poles, two brass 
plumb-bobs, hatchet, and about twelve stakes. 

Location: [See Problem 1.] 

Method : Drive stakes on line, at least one every 100 feet, and also 
one at every point where it is necessary to “break chain,” placing 
them so that an even-foot mark will come on top of the stake. 
Drive the stakes with their broadest dimension in the direction of the 
measurement. Before driving a slake find a point on the grouud 
(using a plumb-bob if necessary) for the location of the stake with 
such accuracy that when the stake is driven the required even-foot 
mark will come somewhere on the top of the stake. Transfer the 
measurements to the tops of the stakes very carefully, using a plumb- 
bob, and mark the point with a pencil. Measure the final fraction 
of a foot to the nearest hundredth by estimating tenths of the grad¬ 
uated tenths of a foot. If only the last foot of the tape is graduated 
into tenths and the last measurement is (e.g.) 37 -f-, then place the 
38-foot mark at the last measuring-point and the fraction of a foot 
may then be read directly at the hub. Measure the distance three 
times. The extreme range of the three values should not be more 
than of the distance. Each student makes a set of three meas¬ 
urements acting as head chainman. The intermediate stakes should 
be pulled and re-driven between each set, so as to make the measure¬ 
ments absolutely independent. Notice the instructions in Problem 1 
in regard to keeping the chain level and making the alignment. 

[right-hand page OF NOTE-BOOK 1 


Oct. 16, 1895 
9:15 a.M. -12:30 p.m. 

A-B-, head chainman 

C-D-, rear “ 

Measurements made with link-chain No. 4 






30 


— 3 - 

LINEAR MEASUREMENTS—TESTING CHAIN (OR TAPE). 

Equipment : Chain (or tape) to be tested, standard tape, spring bal¬ 
ance, two brass plumb-bobs, triangular scale, and a small strip of 
cardboard (e.g., visiting-card). 

Location : A passageway or hall is often the only available place, 
and serves very well, except for interference to and by others. A 
smooth level sidewalk will answer the purpose in fair weather. The 
vacant space uuder the seats of a large grand stand may sometimes be 
utilized as a permanent place for these as well as other long-tape tests. 

1. Set one of the end marks at some well-defined mark on the floor 
(e.g., a scratch in the top of a nail-head) and tack the strip of card¬ 
board under the other end mark of the tape. Pull the tape to a 
tension of 16 lbs., and mark on the cardboard with a sharp hard 
pencil the position of the 100-foot mark. Make a similar meas¬ 
urement with the chain or tape to be tested, and measure the differ¬ 
ence if any to hundredths of an inch, which is then reduced to thou¬ 
sandths of a foot. Make three independent trials. 

2. Observe by three trials for each tension the changes in th ; 
length of the chain (or tape) by making the tension 8 lbs. and then 
20 lbs. Compute the average change for one pound variation in pull. 

3. Hold the chain (or tape) to be tested clear of the floor and note 
what tension is necessary to bring the end marks at the same distance 
apart as when supported throughout its length with a tension of 16 
lbs. Use brass plumb-bobs in this test to bring the end marks over 
the points on the floor. Make three trials. 


Form op Notes. 


Test. 

Stand¬ 

ard 

Tape. 

Tested 

Tape. 

Diff. 

Mean 

Error. 



No. 1 

100.000 

100.017 

100.014 

100.015 

+0.017 

+0.014 

+0.015 

j-+0.015 


Tension 16 lbs. 

No. 2 


0".16 

0".15 

0".16 


0".157 

0".013 
per lb. 

Stretch for change of ten¬ 
sion from 8 lbs. to 20 lbs. 

No. 3 


17 lbs. 
'18 lbs. 
16 lbs. 



17 lbs. 

Required tension, i,e., 
“normal tension.” 



















31 


CHAIN-SURVEYING—LAYING OUT RIGHT ANGLES. 

Equipment: Link-cliain, set of marking-pins, two iron plumb* 
bobs. 

1. Select an area where a square about 250 feet on a side can be 

laid out. Select a and b about 250 feet , a _ m , 

apart. Lay out a right angle at b, using the a 

“3, 4, 5 method.” Measure mb =30 feet. n 

Fix the zero of the chain at b (fastening it 

with a marking-pin) and have one man hold 

the 90-foot mark at m. Swing the 40-foot 

mark until both sections of the tape are taut 

and the angle mbn will then be a right angle. J, --1 0 

Measure off bc=ab. Lay out a right angle 
at c and make cd=ab. Lay out a right angle at d and make da' = ab. 
Measure aa' and compute the “error of closure “ (which equals aa! 
divided by the total perimeter). 

The accuracy of the work will depend largely on having the point 
m exactly on the line ab and on establishing c exactly on the line bn 
produced, and on taking similar precautions at the other corners. 
This accuracy may be promoted, especially on rough ground, by 
sighting with the aid of a plumb-bob line. Similar precautions 
should be taken in the second part of this problem. 

2. To chain a line through an obstacle (imaginary in this 


case). Select two points a 
and b as in figure. Lay off 



/ right angles at b, c, d, and e , 
making be-de. Produce ef 
backward and note value (if 
anything) of bb' and aa'. 
Measure eb. 


c 


Notes.— Sketches as above with statements of the required results 
will constitute a sufficient report. The magnitude of the errors will 
indicate how much reliance can be placed on such methods. 







32 


— 5 — 

CHAIN-SURVEYING—OBLIQUE ANGLES. 




Equipment: Chain, two transit-poles, set of marking-pins. 

Set five marking-pins as witnesses near five designated hubs, which 
should be about 300 feet apart, and form an irregular pentagon. 

1. Measure all the angles by the method 
suggested in the figure, using the formula 

Sm = 2 b' 

If the angle is much greater than 90°', it 
may be more convenient to produce one of the sides and measure the 
supplement of the angle (e.g., angle 3 in the figure). Note whether 
the sum of all the interior angles is nearly 540°. 



Form of Notes. 


Angle 

b 

a 

a 

2b 

\ A 

A 

1 

40 

58.7 

.734 

47° 13' 

94° 26' 

2 

40 

62.8 

.785 

51° 43' 

103° 26' 

3 

40 

29.5 

369 

21° 40' 

(sup.) 136° 40' 

4 

40 

55.2 

.690 

43° 38' 

87° 16' 

5 

40 

68.5 

.856 

58° 52' 

117° 44' 






539° 32' 

Course. 

Bearing. 

Distance. 

Latitude. 

Departure. 



(mend, assumed.) 





1....2 

N 

252.8 

+ 252.8 



2...3 

N 76° 34' E 

204.6 

+ 47.5 

+ 199.0 


3....4 

S 60° 06' E 

320.7 

- 159.9 

+ 278.0 


4... 5 

S 32° 38' W 

207.2 

r- 174.5 

- 111.7 


5 .. 1 

N 85° 06' W 

363.8 

+ 31.1 

-362.5 




1349.1 

+ 331.4 

+ 477.0 





-334.4 

- 474.2 





3.0 

2.8 






































33 


-5 — 

2. Measure the sides. Assume one side as meridian, and from the 
measured angles compute the corresponding bearings of the other 
sides. For example, the bearing of 3 . . . 4 is S 60° 06' E; i.e., the 
bearing of 4 ... 3 is N 60° 06' W. The angle at 4 is 87° 16', which 
means that 4 ... 5 runs in a direction 147° 22 (the sum) from 
North or 32° 38' from South. Therefore the bearing of 4 . . . 5 is 
S 32° 38' W. Compute the latitudes and departures and then the 
error of closure. The latitude equals the distance times the cosine 
of the bearing; the departure equals the distance times the sine of 
the bearing. The error of closure is the quotient (generally ex¬ 
pressed as a fraction whose numerator is one) of the square root of 
the sum of the squares of the errors of the latitudes and of the de¬ 
partures divided by the total perimeter. 



Error of closure = 


2 . 8 » 

1349.1 


_L 

329 


5 










34 




— 6 — 

COMPASS PRACTICE—SIGHTING AND READING 
NEEDLE. 

Equipment: Surveyor’s compass. 

Set up the compass over some fixed hub designated by the in¬ 
structor in charge and point at four well-defined points (e.g., church 
steeples, corners of buildings, etc.), which will be also designated. 
After sighting at all four points in turn and taking the needle read¬ 
ing on each to the nearest 5' of arc, move the declination arc slightly 
and again take the needle readings on all four points. Take four 
such sets of readings. Calling 0 the position of the instrument and 
A, B, G, and D the points sighted at, four values of the angle A OB 
may be obtained by taking the differences of the bearings. Sim¬ 
ilarly four values of each of the other angles may be obtained. The 
discrepancies will give some idea of the reliability of needle readings. 

Form op Notes. 



1st Reading. 

2d Reading. 

3d Reading. 

4th Reading. 

Mean. 

Point A 
“ B 
“ G 
“ D 

N 13° ‘<*0' W 
N 1° 35'E 

N 64° 30'E 

N 77° 45' E 

N 15° 10' W 
N 0° 20' W 
N 62° 35' E 

N 76° 0'E 

N 16° 25'W 
N 1° 30' W 
N 61° 15'E 

N 74° 55' E 

N 17° 10' W 
N 2° 15' W 
N 60° 45' E 

N 74° 15' E 


Angle AOB 
“ BOO 
“ COD 

14° 55' 

62° 55' 

13° 15' 

14° 50' 

62° 55' 

13° 25' 

14° 55' 

62° 45 

13° 40' 

14° 55' 

63° 00' 

13° 30' 

14° 54' 
62° 54' 
13° 27' 

























Abnormal discrepancies in the angles may sometimes occur on 
account of the attraction caused by a knife, l^eys, or a steel key- 
chain in the pockets of the observer, or by a steel band in the brim 
of the observer’s hat. A wagon, temporarily halted near the instru¬ 
ment, especially if loaded with iron, will probably cause an appre¬ 
ciable disturbance of the needle. The student will find it a useful 
exercise, in connection with this problem, to determine the effect, 
by careful and repeated trials, of purposely allowing such sources 
of attraction to be held first on one side and then on the other side 
of the instrument. To be of any value, this test should only be 
made in a way that might occur naturally while using the instru¬ 
ment. 


Tower Blind Asylum. 

S W. cor. 34th and Market Sts. 
Tower of The Bartram. 

N. W. cor. Chem. Lab. 






36 


— 7 — 

COMPASS PRACTICE—ERRORS OF SIGHTING ; SETTING 
BY NEEDLE AND LINING IN. 

Equipment: Compass, transit-pole, foot-rule, board about 2 feet 
long and 6 or 8 inches wide. 

1. Error of sighting. Set up the compass where a clear view of 
600 feet may be had. Have the rodman pace off 200 feet and lay the 
board on the ground with its length transverse to the line of sight, 
takiug care to fix the board very firmly. If a bit of smooth clean 
pavement is at hand, it will be preferable to the board, being rigid. 
Sight at the rod held at some marked point near the centre of the 
board. Clamp the vertical axis. Have the rodman measure'the 
distance of this point from one end of the board and then move the 

Form of Notes. 


Distance. 

Distance of 
rod from end 
of board. 

Error. 

id) 




200 

// 

12.0 

12.8 

12.6 

10.8 

etc. 

// 

.18 

.62 

.42 

1.38 

etc. 

.0324 
.3844 
.1764 
1.9044 
etc. . 


• 

Sum = 
Mean = 

121.8 

12.18 

• 

5.2680 



600 

(simi 

larly) 




200 

(simi 

larly) 




600 

(simi 

larly) 































87 


rod away. Line in the rod again, and again have the rodman 
measure its position. Repeat at least 10 times. Compute the 
“probable error ” (see Appendix A) of a single sighting in linear dis¬ 
tance. Reduce this error to its angular value by dividing the error 
by the distance from the instrument. The quotient is the tangent of 
the angular error. 

Repeat this test at a distance of 600 feet. 

2. Error of setting by needle and lining in. Read the needle 
to the nearest 5' when pointing at some one point on the board. 
Turn the compass a little on its vertical axis and have the rodman. 
move the rod away. Set the compass at the same needle reading, and 
again line in the rod. Have the rodman note the position of the rod 
as in the first part of the problem. Repeat 10 times at both 200 and 
600 feet distance. 

Caution. —Be sure that the declination arc is not disturbed during 
the progress of the work. 



= .000215 = tan 0° 0' 44" 


Constant compass reading—N 62° 25' E 


Constant compass reading—N 67° 50' E 











COMPASS PRACTICE-RUNNING OUT A LINE HAYING 
A CONSTANT BEARING. 

Equipment: Compass, transit-pole, set of marking-pins. 

Run out a line about £ mile long, on uneven ground, setting pins 
at each station, using a constant bearing. Pace the distances between 
stations. Then run back by the reverse bearings and have the rod- 
man note the discrepancy at each station. After noting the error the 
rodman transfers the marking pin to the new position, which will 
make the entire return line continuous and independent. Be careful 
to select stations where there will be no local attraction {e g., from 
lamp-posts, iron fences, etc.). Where local attraction is suspected, 
take a back-sight and determine its amount. The work will demon¬ 
strate the compensating character of the errors, the degree of 
accuracy that may be expected, and the frequency even in suburban 
localities of greater or less local attraction. 

Form of Notes. 


Sta. 

Bearing. 

Distance. 

Errors 
on return. 


0 

S 65J4° W 

350 

S 0.4 


1 

S 65^° W 

420 

NO.6 


2 

S 60^° W 

370 

N 1.8 

Local attr. suspected at sta. 2. B. S. 





to sta. 1, N 60E. 

3 

S 65J4° W 

640 

N 0.4 

B. S. to sta. 2, N 65^j E. 

4 

S 65J4 0 W 

260 

S 1.0 


5 

S 65^° W 

480 

S 0.3 


6 


370 


Bearing on return line from sta. 6, 





N 65H E. 


















39 


» 


— 9 — 

LEVEL PRACTICE—READING ROD AND SETTING , 
TARGET. 

Equipment: Wye-level, “Philadelphia” level-rod. 

1. Send the rodman about 200 feet away on a nearly level stretch. 
Sight on the rod held on a peg or on a smooth solid knob of stone or 
brick and direct the setting of the target, which is then read to the 
nearest .001 foot and recorded by the rodman. The rodman then 
moves the target a few inches up or down and the target is again set 
and read. Obtain 10 such values. Find the “probable error” (see 
Appendix A) of the mean and of a single observation. 

2. Turn the rod so that the leveller sees the numbers on the back 
of the rod. The leveller reads the rod directly to the nearest .01 
foot and records the reading. The rodman then extends the rod 
about 0.1 foot, reading and recording the exact movement to the 
nearest .01 foot as shown by the index. The leveller again reads 
and records the rod reading. This is done 10 times. Subtract the 
total rod movement as reported afterward by the rodman from the 
rod reading to which each corresponds. The results should be 
nearly identical. Compute the “probable error” as in Part 1. This 
exercise tests the ability to read a “ speaking rod.” 

3. Repeat Part 1 at a distance of about 600 feet. This will test 
the ability to read a target at a distance. 

Form of Notes. 


No. of 

Rod 

d 

d 2 

Dist. 



observ. 

reading. 



1 

5.203 

.001 

.000 001 

200 



2 

5.206 

.002 

004 




3 

5.199 

.005 

025 




etc. 

etc. 

etc. 

etc. 




io‘ 

5.‘ 201 

!oo3 

.000 009 



/.000156 

E x - .6745 V 1Q _ 1 - = .0028 

Sum 

Mean 

52.041 

5.204 


.000156 



.0028 

E m = —= .00088 
m Vio 

(simi 

larly) 



600 



No. of 

Rod 

Rod 

Diff. 

d 

d 2 


observ. 

reading. 

extension. 


1 

8.63 

_ 

8.63 

.02 

.0004 

Distance about 200 feet. 

2 

8.74 

.08 

8.62 

.03 

9 


3 

8.81 " 

.19 

8.62 

.03 

9 


4 

8.96 

.30 

8.66 

.01 

.0001 


etc. 

etc. 

etc. 

etc. 

etc. 

etc. 







































40 


— 10 — 

LEVEL PRACTICE—DIFFERENTIAL LEVELLING. 

Equipment: Level, level-rod. 

Location: Two bubs should be cbosen, about £ to \ mile apart, 
and between wbicb tbe natural slope is quite considerable, allowing 
an extreme difference of elevation of 50 feet or more. 

Method: The problem is to find tbe difference of elevation of tbe 
two bubs, tbe elevations of intermediate points not being desired. 
Set up tbe level so that a reading may be taken on tbe level-rod when 
it is held on tbe first hub. To reduce the number of “set-ups” as 
much as possible, tbe level should be set up as far away from the 
bub as possible (up to a limit of about 300 feet) and yet have it 
possible to read tbe rod set on the bub. If tbe natural slope is very 
steep, it may be difficult to make this distance much greater than 10 
feet, which is about tbe lower limit of distinct focussing with tbe 
telescope. Tbe leveller reads tbe rod to tbe nearest hundredth of a 
foot—not using tbe target. [This assumes tbe use of a “Philadelphia” 
rod.] This rod reading is recorded as a “back-sight”—B. S. Then 
the rodman finds a suitable “turning-point” (T. P.), which should 
preferably be as far from tbe level as tbe level is from tbe hub. The 
turning-point may be any firm object which has a flat or convex 
upper surface (e.g., a protruding point of firm well-bedded stone). 
Tbe reading on tbe turning-point is recorded as a “ fore-sight ”—F. S. 
Then tbe instrument is moved as far ahead of tbe turning-point as 
possible and a back-sight is taken on that turning-point. These 
operations are repeated until it is possible to take a fore-sight on tbe 
final bub. Tbe sum of tbe back-sights minus tbe sum of tbe fore¬ 
sights gives tbe difference of elevation. If tbe difference is positive, 
the first bub is lower than the second and vice versa. If tbe bubs are 
on opposite sides of a valley and at nearly tbe same elevation, it may 
require a rigid application of tbe above rule to determine which is 
the higher point. 

Form of Notes. 

[The Form of Notes should be identical with that given for 
Problem 11 except that there will be no “intermediate sights ” (I. S.), 
all rod readings being recorded either as “fore-sights” (F. S.) or 
“back-sights” (B. S.).] 


41 


— 11 — 

LEVEL PRACTICE—PROFILE LEVELLING. 

Equipment: Level, level-rod, 50-foot (or 100-foot) tape. The 
student must provide himself with a sheet of profile-paper (Plate 
A). 

Location: If a line of points spaced 50 or 100 feet apart is already 
located, they may be utilized for this problem. In city streets a 
line of points 50 or 100 feet apart and extending for } or | mile along 
the curb-line or centre line may be quickly put in, the points being 
marked on the pavement with blackboard crayon. In the country 
the points may be marked with marking-pins, which are aligned by 
eye. 

Method: Assume some suitable point (e.g., a permanent hub on the 
surveying-ground) near one end of the line of points as a “bench¬ 
mark”—B. M. Assume the datum-plane to be 100 feet below this 
B. M. Set up the instrument (as in Problem 10) so as to take a back¬ 
sight on the bench-mark. Then take a rod reading on as many of 
the line-points as may be obtained from that setting of the level. 
These readings are called “intermediate sights”—I. S. Then take a 
fore sight on a suitable turning-point and move the level ahead (as in 
Problem 10), continuing the work similarly until the elevation of all 
the points is obtained. Finally take a reading on some suitable 
bench-mark near the other end of the line of points. 

Draw the profile on profile-paper (Plate A), using as horizontal 
scale 100 feet to the inch, and as vertical scale 10 feet to the inch. 


Form of Notes. 


Sta. 

B. S. 

H. S 

F. S. 

I. S. 

Elev. 


B. M. 

4.67 

104.67 



100.00 

B. M. on top of fire-plug at 

A 0 




6.82 

97.85 

corner of M. and N. 

+ 50 




5.34 

99.33 

streets. 

A 1 




3.86 

100.81 


+ 50 




2.38 

102.29 


T. P. 



1.46 


103.21 

• 


9.65 

112.86 





A 2 




9.06 

103.80 


etc. 




etc. 

etc. 















42 


- 12 - 

level PRACTICE—TESTING ADJUSTMENTS. 

Equipment: Wye-level, level-rod, 100-foot tape, set of marking- 
pins. 

The object of this problem is to make the student as familiar as 
possible with the adjustments, without subjecting the instrument to 
the injury that lyould result from excessive disturbance of the 
adjusting-screws, especially in unskilful hands. Therefore no 
ad justing-screws are to be disturbed, but the value of each error of 
adjustment is determined, assuming all previous adjustments as 
already made. Making three trials of each adjustment will demon¬ 
strate to the student that unless extreme care is taken in handling the 
instrument, the three trials will not exactly agree, and that he will 
sometimes find an apparent small error when the adjustment is prac¬ 
tically perfect. This note applies equally to Problem 16. 

1. To make the line of sight parallel to the axis of the 
bubble. 


Form of Notes. 


Adjustment 


1st Trial. 

2d Trial. 

3d Trial. 

Mean. 


Tel. direct 

3.684 

3.986 

3.902 


la 

Tel. inverted 

3.692 

3.995 

3.909 


^ diff. 

0.004 

0.0045 

0.0035 

0 0010 


Eye end 

17.6 

17.7 

17.8 


lb 

Object end 

16.0 

16.0 

16.1 

0.83 div. 

^4 diff. 

0.8 

0.85 

0.85 

2 







Eye end 

16.2 

16.1 

16.1 


3 

Object end 

17.7 

17.7 

17.5 

0.75 div. 


>4 diff. 

0.75 

0.8 

0.7 



15.8 

15.4 

14.8 



E \ 

3.6 

5.6 

4.2 



l 

12.2 

9.8 

10.6 




15.2 

13.2 

13.1 


Angdlar 

° 1 

2.8 

3.1 

2.2 


value of one 
division of 

12.4 

10.1 

10.9 


level-tube 


4.931 

4.867 

4.831 



* 1 

4.211 

4 294 

4.208 



0.720 

0.573 

0.623 



a" 

30". 2 

29".7 

29". 9 

29". 93 






























43 


a. Measure off a base 400 feet long on nearly level ground and set 
up the level at one end of the line. Sight at the rod at the other 
end. Have the vertical axis clamped, and the whole instrument as 
rigid as possible except the clips over the telescope, which should be 
loose so that the telescope can be rotated in the wyes without shaking 
the instrument. If the line of collimation is in adjustment the rod 
reading will be the same, when the telescope is rotated half-way, as 
at first. The difference, if any, is twice the error. Make three tests. 
Raise or lower the telescope between each test so as to bring the 
readings on a different part of the rod. 

b. Bring the telescope directly over a pair of levelling-sCrews and 
clamp the vertical axis. Bring the bubble to the centre with the 
levelling-screws. Reverse the telescope in the wyes. If the bubble 
again comes to the centre, the wyes are level and the line of collima¬ 
tion is parallel to the axis of the bubble. If the bubble does not 
come to the.centre, move the levelling-screws until the bubble is 
moved half-way back. Reverse the telescope again, and the bubble 
will come very nearly if not quite to its position before reversal. 
Keep adjusting the levelling-screws until the bubble remains in one 


Dist. = 400 feet. Rod-readings are less when telescope is direct; therefore line 

004 

of collimation points down ^r- = .00001 = tan 0° 0' 02". 


Eye end high; therefore telescope points down 0.83 div. of bubble. 

Bubble moves forward about one division when swung to the right about 30°; 
moves backward when swung to the left. 


Dist. = 400 feet. 









44 


place in the tube for the telescope in either position. Under those 
conditions the wyes are level and distance of the centre of the bubble 
from the centre of the tube is the error of adjustment. Record this 
distance measured in divisions of the bubble-tube. Make three tests. 
Make t e determinations independent by purposely throwing the 
instrument out of level between each test. 

2. To bring the bubble axis into the vertical plane through 
the axis of the telescope. Bring the bubble to the centre of the 
tube, first observing that the bubble-tube hangs directly under the 
telescope. Rotate the telescope slightly in the wyes, and note 
whether the bubble remains in the centre, or if it moves in one direc¬ 
tion for a swing to the light and in the other direction for a swing 
to the left. 

3. To make the axis of the wyes perpendicular to the ver¬ 
tical axis of the instrument. Bring the bubble to the centre 
over both pairs of levelling-screws and then rotate the instrument 
about its vertical axis 180°. Correct one half the error, if any, with 
the levelling-screws ; make the same correction for the other level¬ 
ling screws. The bubble will then remain nearly, if not quite, in 
the same place for any position of the telescope. Keep adjusting 
until this is true. The distance of the centre of the bubble from the 
centre of the tube is a measure of the correction that should be made 
by the adjusting-screw under one of the wyes. Record the error, 
if any, in divisions of the bubble-tube. Make three tests by purposely 
throwing the instrument out of level after each test and making an 
independent determination. 

Value in seconds of arc of one division of the level-tube. 
Run the bubble to near one end of the tube, but still keeping both 
ends of the bubble within the graduated divisions on the tube. 
Take a readiug on the rod held 400 feet away. Move the bubble to 
the other end of the tube and again read the rod. 

Let E= the movement, in divisions, of the eye end of the bubble ; 

Let 0 = the movement, in divisions, of the object end of the 
bubble ; 

Let R = difference in rod readings ; 

Let a" — _—_. 

400 sin 1" 

z 

Obtain three values. 




45 


- 13 - 

transit PRACTICE—ANGLE MEASUREMENTS. 

Equipment: Transit. 

•Set up the transit over a designated hub and point at two well- 
defined points (edge of a cornice seen against the sky or a church 
spire) which will be also designated. With vernier A at any reading 
(i.e., not 0°) set the vertical cross hair on the left-hand point. Read 
the degrees and minutes of vernier A and the minutes of vernier B. 
Take the mean of the readings of minutes. That, with the degrees 
of vernier A, constitutes the “mean.” Loosen the upper plate, 
point at the other mark, and clamp again. Again take the mean 
reading of the verniers. The difference of the means is the liori 
zontal angle. With the upper plate clamped and lower plate loose, 
turn back to and sight on the left-hand mark and clamp the lower 
plate. Loosen the upper plate ; point at the right-hand mark; clamp 
the upper plate ; take the mean of the vernier readings. The differ¬ 
ence between this mean and the previous mean should be practically 
the same as before. Continue this method until ten such values of 
the angle are obtained. They should not differ more than 1' of arc. 
Estimate each vernier reading to half-minutes. Compute the “ prob¬ 
able error” of thelnean and of a single value of the angle. 


Form of Notes. 


Vern. A. 

Vern. B. 

Mean. 

Diff. 

d 

d 2 


20° 16' 0" 

15' 30" 

20° 15' 45" 


_ 


Left-hand mark, church 





steeple, cor. A.& B. Sts. 

74° 18' 

18' 

O 

QO 

54° 02' 15* 

6" 

36 

Right-hand mark, tower 




finial, cor. M. & N. Sts. 

128° 20' 30" 

21' 

128° 20' 45* 

54° 01' 45* 

24 

576 


182° 22' 

22' 30" 

182° 22' 15* 

54° 01' 30" 

39 

1521 


236° 24' 30" 

24' 

236° 24' 15* 

54° 02' 

9 

81 

7 5264 

^ = .6745 V Hh=~l = l 6 " 3 







16.3 

etc. 

etc. 

etc. 

etc. 

etc. 

etc. 

Em= —= = 5".l 

V10 




21' 30* 


5261 




Mean 

54° 02' 09" 


























46 


- 14 - 

transit PRACTICE-TRAVERSING. 


Equipment: Transit, transit pole, set of marking-pins. 

Set five marking-pins as witnesses near five designated hubs. Set 
up the transit over one hub ; set vernier A at zero ; loosen the needle 
and revolve the whole instrument until the telescope points south, ns 
shown by the needle. Clamp the lower plate ; loosen the upper plate 
and point at the next hub ; read vernier A and the needle. Set up 
the transit at the next hub ; plunge the telescope and point at tbe 
previous station •, test vernier A to see that the plate has not slipped ; 
read the needle ; plunge the telescope again to direct position and 
turn to the third hub and again read vernier A and needle. Take 
observations similarly at hubs 3, 4, and 5 ; when at station 5 take a 
fore-sight to station 1. Finally, reoccupy station 1, sighting back at 
station 5 for a back-sight; the reading of vernier A for the fore sight 
to station 2 should exactly agree with the former reading. A differ¬ 
ence of more than a very few minutes of arc indicates gross inaccu¬ 
racy or a blunder. The object of the needle readings is to guard 
against blunders. As soon as the vernier and needle readings are 
recorded, note whether the azimuth and needle reading agree. For 
example, an azimuth of 217° 42' signifies a bearing of N 37° 42' E, 
and the needle reading should be as near this as it may be read, un¬ 
less there is local attraction. The degrees on the horizontal plate 
should be read from the row that is continuous to 360°. 

Form of Notes. 


Inst, at 

Pointing to 

Vern A 

Needle. 



Sta. 1 

South 

0° 

South 




a2 

274° 03' 

S 86° E 



A 2 

A 1 

274° 03' 

S 85° 50' E 


Tel. rev. 


a3 

192° 16' 

N 12° 20' E 



A 3 

a2 




Tel. rev. 


a4 


.... 



etc. 





















47 


— 15 - 

transit PRACTICE—ROD ALIGNMENT; SETTING 
ALIDADE. 

Equipment: Transit, transit-pole, board about two feet long and 
six or eight inches wide, foot-rule. 

1. Send the rodman to a point about 300 feet distant; have him 
lay the board on the grouud where it will be firm, with its length 
perpendicular to a line to the transit; have him set the rod on the 
board (near the middle) and measure and record the distance to one 
end of the board (measuring in inches and tenths). The transitman 
sights at the rod, clamping both plates. The rodman moves the rod 
away and is again lined in by the transitman. The rodman again 
measures and records the distance to the end of the board. Obtain 
10 such values. Compute the “ probable error” of a single sighting 
and of the mean of all the sightings. Reduce the probable error of 

the mean to its angular value = tan 

2. The transitman reads vernier A when the transit points at some 
one setting of the rod, and between each subsequent setting unclamps 
and moves slightly the alidade, then resets the alidade at the same 
reading and again lines in the rod as in Part 1. Obtain 10 such 
values. Compute the probable error and the angular value of the 
error as before (Part 1). 

3. Repeat both of these tests at a distance of about 1000 feet, 
which may be measured by pacing. 


Form of Notes. 


Dist. 

No. of 
Rod Setting. 

Dist. of Rod 
from End 
of Board. 

d 

a 2 



300 

1 

2 

3 

etc. 

’io’ 

Mean 

12.2 

12.5 

12.7 

etc. 

li'.s 

.04 

.34 

.54 

etc. 

i36 

.0016 

.1156 

.2916 

etc. 

!l296 


/1.6240 

Ei = .6745 V io3T * • 286 " 
= .024' 

.286 

-non" — 

121.6 

12.16 

1.6240 

-H/7TI — /-ja — , — *UuiO 

.0075 

= .000025 = tan 0° O' 05" 

300 

1 

2 

.... 



Vern. A 
30° 27' 






















48 


- 16 - 

transit PRACTICE—TESTING ADJUSTMENTS.* 

Equipment: Transit, transit-pole, level-rod, 100' steel-tape, foot- 
rule. 

1. To make the plane of the plate bubbles perpendicular 
to the vertical axis. Set up the transit and make the plate bubbles 
parallel to opposite levelling-screws. Bring both bubbles to the 
centre ; revolve exactly 180°. The distance (measured in divisions 
on the bubble-tube) of the centre of the bubble from the centre of 
the scale is twice the error. With the levelling-screws throw the 
transit out of level and begin over again so as to make an absolutely 
independent redetermination. Obtain three independent values for 
the error of this adjustment. 

2. To make the line of sight perpendicular to the horizontal 
axis of the telescope. Set the transit where a clear sight of about 
300 feet each way in opposite directions may be obtained. Point the 
telescope at some well-defined point (e.g., the corner of a building 
having a good background). Plunge the telescope and line in the 
rodrnan at some poiut-about 300 feet away. Revolve the alidade and 
point at the first point selected. Again plunge the telescope and line 
in the rodman for a point beside the one previously set. The distauce 
between these two points is four times the error. Make three tests. 

3. To make the horizontal axis of the telescope perpen¬ 
dicular to the vertical axis of the transit. Set up the transit 
about 20 feet from the vertical wall of a building. Select a point 
about as high on the wall as the telescope may be made to point at. 
Swing the telescope down and have the rodman make a mark on the 
wall down near the ground. Plunge the telescope, revolve the in¬ 
strument 180°, and again point at the upper point. Swing the tele¬ 
scope down and have the rodman make another mark beside the first 
lower mark. A vertical plane through the instrument and the upper 
mark will pass midway between the two lower marks. Record the. 
distance of the instrument from the wall, the vertical distance (esti¬ 
mated) between the upper and lower points and the distance between 
the two lower points. Make three tests. 

4. To make the axis of the telescope bubble parallel to the 
line of sight. Level the telescope and mark a point on the ground 
directly under the eyepiece (driving a peg if necessary). Measure 


* See first paragraph of Problem 12, page 16. 



49 


tlie height of the centre of the eyepiece above the point with the 
level-rod ; call the reading a. Establish another point about 300 
feet away and at about the same level. Read the level rod placed 
on it ; call the reading b. Transfer the instrument, to the second 
point and set it up so that the eyepiece is directly over the point. 
Measuie the height of the eye-piece above the point, calling it c, and 
also read the level-rod held on the first point, calling it d. Then we 
should have d — a = c — b if the instrument is in adjustment. If 

m, 

not, then (d — a) — (c — b) = 2m. ( |^ staDC g = tan If m is positive 

the line of collimation points upward. Make three tests. 

5. To make the vernier of the vertical circle read zero 
when the line of sight is horizontal. This test is only applicable 
to instruments having fixed vertical circles. As an adjustment it is 
only applicable to instruments having adjustable verniers. Make 
the telescope level by means of the level bubble and note the amount 
and direction of the “index error.” Make three tests by turning the 
telescope out of level between each test. 

Best location for these tests. Tests 2 and 4 require an unob¬ 
structed, nearly level stretch of 300 feet in one direction, and Test 2 
a suitable point about 300 feet in the opposite direction. Test 3 
requires a high wall about 20 feet from the instrument, and Tests 1 
and 5 may be made anywhere. When an unobstructed line of suffi¬ 
cient length can be found, parallel to a wall and about 20 feet away, 
a single setting of the instrument will serve for all the tests. 


Form of Notes. 


Test. 

, 1st Trial. 

2d Trial. 

3d Trial. 

Mean. 



1 j 

N .3 div. 

N .5 div. 

N .4 div. 

iN .40 div. 


Bub. par. to tel. 

1 1 

E 1.2 “ 

E 1.0 “ 

E 1.3 “ 

E 1.17 “ 

1.46 

—= 0.365 

“ perp. “ “ 

2 


M" 

m" 

1".46 

dist. about 350 ft 

3 

t*" 

%” 

J4" 

it" 

5 5 

16 _f ' 8 = 32 

wall 24 ft. from 
transit; upper 

4 a 
b 
c 

4.26 

3.78 

4.61 

4.51 

4.04 

4.61 

4.51 

4.02 

4.47 


dist. = 364 

and lower 

points about 
40 ft. apart. 

d 

5.13 

5.14 

5.03 




m 

.02 

.03 

.035 




tan a 1 

.000 054 

.000 081 

.000 094 

+ 0° 0' 15" 



a 

4-0° 0' 11" 

+ 0° O' 16", 

4-0° O' 19" 



5 1 

4-0° 3' 

4- 0° 3J4' 

+ 0° 2y 2 ' 

4- 0° 03' 

























50 


- 17 - 

transit PRACTICE-STADIA READINGS. 

Equipment: Transit, stadia-rod, 5-foot rod (a light rod, graduated 
in feet and tenths, for obtaining H. I). 

Send the rodman to a point about 300 feet away. Note the height 
of the horizontal axis of the telescope above the ground. Measure 
the “/-f- c.” Point at the rod with the middle cross-wire at a read¬ 
ing on the rod equal to the height of the axis. Read the upper, 
middle, and lower cross-hairs, also the vertical arc. Note the 
equality of the upper and lower wire interval. For a check on the 
distance and as additional practice, move the tangent-screw to the 
vertical arc very slightly, and reread the three cross-hairs. Take six 
such readings. [Although by strict theory these wire intervals 
should differ slightly with any change in the vertical angle, there 
should be no appreciable variation in them for slight changes, 
especially if the vertical angle itself is small. For example, with a 
vertical angle of 10° and a distance of 1000 feet, the vertical angle 
would have to be changed 10' in arc to make a difference of 1 foot 
in the apparent horizontal distance, and 10' of arc in 1000 feet would 
mean a movement of 2.9 feet of the wire over the rod.] Compute 
the true horizontal distance and the difference of elevation for each 
observation taken. The horizontal distances should agree (to the 
nearest foot) and the “differences of elevation” should vary by the 
differences in the readings of the middle wire. Also observe the 
differences in the wire readings, i.e., the wire intervals. 

Repeat these observations at about 600 feet and at about 1000 feet. 
These observations will test the ability to do accurate work at these 
distances. 

Form of Notes. 


Approx. 

Dist. 

Wire 

Read¬ 

ings. 

Vertical 

Angle. 

Horizontal 

Distance. 

Diff. 

Elev. 

Wire 

Inter¬ 

vals. 


300 

(6.24 
-<4.80 
( 3.36 

- 2° 24' 

289 

- 12.11 

j 1.44 
\ 1.44 

H. I. = 4.80 
f+c= 1.6 

300 

(6.32 

-<4.88 

(,3.43 

— 2° 23' 

289 

eo 

© 

oi 

1 

J 1.44 
\ 1.45 


etc. 

etc. 

etc. 

etc. 

etc. 

etc. 


















51 


- 18 - 

TRANSIT PRACTICE—TRAVERSING WITH STADIA. 

Equipment: Transit, stadia-rod, set of marking-pins, 5-foot rod. 

Make a traverse as in Problem 14, the distances between stations 
being measured by stadia both while taking fore-sights and back¬ 
sights. Unless the transit has a full vertical circle the vertical angle 
for the back-sight cannot be read with the telescope plunged. It is 
therefore necessary with most instruments when taking a back-sight 
to set vernier A at a reading exactly 180° different from the reading 
for fore-sight, leaving the telescope in direct position. The needle 
reading should also be 180° different from its reading for fore-sight. 
Always direct the middle wire to a point on the rod at the same 
height as H. I. to obtain the vertical angle. The horizontal distance 
may be most quickly read by raising or lowering the telescope until 
the lower cross-wire comes on the nearest even footmark and then 
reading the upper wire. As shown in the previous problem, this in¬ 
volves no appreciable error. Check the horizontal angles by the 
closure of the last course. Also see whether the algebraic sum of 
the differences of elevation equals zero. Compute the true horizontal 
distance to the nearest even foot, the vertical difference of elevation 
to the nearest even tenth of a foot. 


Form of Notes. 


Sta. 

Vern.A 

Needle. 

Vert. 

Ang. 

Wire 

Interval. 

Hor.Dist. 

Diff 

Elev. 

Inst, at a 1 

0° 

S 






A 2 

20° 16' 

S 20° 15' W 

- 2° 16' 

1.80 


) 







1 

>-181 

-7.1 

Inst, at a 2 
B S. a1 

200° 16' 

N 20° 15' E 

- 2° 15 

1.80 

! 



a 3 

118° 47' 

N 61° 1C'W 

-1- 1° 22' 

1.18 


! 


Inst, at A 3 








etc. 






























52 

— 19 - 

plane-table PRACTICE. 

Equipment: Plane table (including alidade, table-level, and com¬ 
pass-box), stadia-rod, steel tape, set of markiug-pius. Studeuts must 
each provide a sheet of drawing-paper about 16" X 24" (detail paper 
will answer the purpose), scale, hard sharp pencil, and rubber. 

Measure a base-line ( AB) about 300 feet long. Set up the plane 
table at A. Orient the table and draw in the base-line (30 feet to an 
inch) so that the proposed sketch will come properly on the paper. 
Draw a magnetic meridian through A, using the compass-box. 
Draw lines toward several points (trees, building corners, etc.), which 
are visible from both A and B and, determining their distances by 
stadia, plot them at once to scale. 

Set up the table at B; orient by sighting to A ; check the orienta¬ 
tion with the compass; check the plotting of the other points by 
sighting to them. Sight to some station point G, visible from both 
A and B, drawing an indefinite line toward it. 

Set up the table at G; orient by sighting to B ; check the orienta¬ 
tion with the compass ; locate G on the drawing by sighting to A. 
Check the plotting of G by sighting at some of the other points 
already located. 

Notes. —In the note-book record date, time of work, number of 
problem, and personnel of party. The drawing, properly lettered, 
should be handed in with the note-book. Each student of the party 
may plot his own sheet for each station while the instrument is set 
there—thus saving time. When another sheet is placed on the table 
at any station, it simply requires a readjustment of the orientation. 
The accuracy is not necessarily impaired. Extreme accuracy in 
drawing is essential for good results. 

Alternative problem. Since the amount of time which any one 
student may be permitted to devote to this problem is generally lim¬ 
ited, it is'sometimes preferable to make a somewhat extended survey 
involving frequent repetitions of the above principles, and in turn 
assigning several parties of students to the problem. The base-line 
should then be increased to 500 feet or more and the scale reduced 
to 100 or even 200 feet per inch. A detailed topographical survey 
may thus be made of a considerable area, which will be a more in¬ 
teresting problem than the solution of abstract mathematical prin¬ 
ciples. 


— 20 — 

PANTOGRAPH. 


53 


Equipment: Pantograph, sheet of drawing- paper (detail paper will 
suffice) about 18" X 24", two triangles, scale, hard pencil. All 
equipment except the pantograph should he furnished by the 
student. 

Use extreme care not to strain the arms or clamps while altering 
the ratios of the arm lengths. Record in each case by a sketch the 
arrangement of the fixed point, tracing point and pencil, and also the 
distances (measured to the nearest .01") of these points from the 
pivots. 

1. Set the instrument so that it will magnify 3.5 diameters and 
prove the work by making an enlarged copy of a 2" square, and 
also of a 2" circle. 

2. Set the instrument so that it will reduce 2.5 diameters and 
prove the work by making a reduced copy of a 5' square and also 
of a 5" circle. 

3. Set the instrument to retrace at the same scale any given figuie 
and prove by applying it to some irregular figure sketched on the 

Sll66t. 

Notes. —The drawing, with all results suitably recorded, will 
constitute the report. In the field-book record simply date, num¬ 


ber of problem, etc. 

Principles involved. Let F represent the fixed point, T the tracer, 
and P the pencil or marker. F, T , and P 
must always be in a straight line. FcT and 
FbP must always be similar triangles ; in 
this instrument they are also isosceles tri¬ 
angles. abcT is always a parallelogram. 

The ratio PF : TF is the ratio of enlarge¬ 
ment. To reduce, transpose P and T. To 

retrace at same scale, make the parallelogram equilateral and trans¬ 
pose F and T. The retraced figure will then be inverted. 



54 


— 21 — 

POLAR PLANIMETER. 

Equipment: Planimeter, sheet of drawing-paper about 18" X 18", 
two triangles, scale, drawing-compasses, hard pencil. All equip¬ 
ment except the planimeter should be furnished by the student. 

Draw on the paper, with extreme accuracy , a circle 2" in diameter, 
a circle 6" in diameter, a 2" square and a 4" square. The 6" circle 
should be nearly in the centre of the sheet, so that when the fixed 
point is placed inside the circle and the pointer traced around the 
circumference, the rolling wheel shall not run off the paper. Set 
the index of the arm of the planimeter at 295.5. With the pointer 
on some definite point of the perimeter of one of the figures read an 
initial reading ( not necessarily zero) on the wheel and vernier. 
Move the pointer around the perimeter, turning to the right. When 
the starting-point is again reached, take the final reading. The 
cifference in the readings divided by the known area in square 
inches gives the reading per square inch. Compute the quotient to 
three significant figures. Trace each figure similarly twice. The 
readings per square inch on all the figures should agree to a small 
fraction of one per cent. 

Obtain the area of the “ zero-circle ” by taking the mean of two 
readings of the six-inch circle with the fixed point inside the circle, 
and then taking the algebraic diference of the mean readings (with 
the fixed point first inside, then outside), which difference, divided 
by the average reading per square inch, gives the area of the zero- 
circle in square inches. In taking the reading with the fixed point 
inside the circle, and the pointer turning as usual to the right, the 
index wheel will run backward, giving a negative quantity. Watch 
the index wheel carefully to note the whole number of revolutions. 
A simple way is to add, say, 30 to the reading, making it, say, 34.769 ; 
watch the wheel run backward through 0, and the readings are then 
in the twenties. Following it backward similarly until the pointer 
has again arrived at the starting-point of the perimeter and the read¬ 
ing, say 8.462, can be subtracted from 34.769, thus giving directly 


55 

the total number of backward revolutions of the rolling wheel. This 
number is negative. The number obtained when the fixed point is 
outside the circle is positive. Therefore their algebraic difference 
is their numerical sum. 

To compute the circumference of the rolling wheel. Use 
the formula A = Inc, in which c = the required circumference, n = 

the mean record of revolutions per square inch, l = 

147.75mm = 5.817 inches, A — 1 square inch. 

Check on area of irregular figure. Draw a ‘ ‘ free-hand ” figure 
enclosing an area, the figure being purposely made very irregular. 
Obtain its area by tracing the pointer around the perimeter, and divid¬ 
ing the difference of the initial and final readings by the previously 
obtained average reading for one square inch. The quotient is the 
area in square inches. Check this result by carefully cutting out the 
irregular figure and also one of the squares and weighing the pieces 
on a chemical balance. With a drawing-paper of uniform thickness 
and texture the weights should be proportional to the areas. 

Notes. —The drawing, with all results suitably recorded on it, 
will constitute the report. In the field-book record simply the 
dale, number of problem, etc. 


295.5 j wm 



56 




— 22 — 

BAROMETRIC LEVELLING 

Equipment: Mercurial and aneroid barometers, pocket thermom¬ 
eter. Each student must be provided with a watch, which should 
i>e compared before beginning work, and a small sheet of profile- 
paper (Plate A). 

Use of the Mercurial Barometer. The mercurial barometer is 
to be observed in its fixed position in the office, (a) Read the 
attached thermometer; this should be done first, as the heat of the 
observer’s body will be sufficient to alter the reading in a short time. 
(b) Bring the surface of the mercury in the cistern to the tip of the 
ivory index. This is most accurately done by raising the mercury 
(by means of the large screw at the bottom) until the tip dips slightly 
into the mercury and forms a slight depression ; lower the mercury 
until, as seen by the reflection of light on the surface of the mercury, 
the depression just disappears. A magnifier will be of assistance in 
doing this. Tap the barometer very gently to destroy the adherence 

Form of Notes. 


Mercurial Barometer Readings. 


Time. 

Barom. 

Att’d 

Therm. 

Red. to 32°. 

Temp, of 
External 
Air. 

Corr. 

Reading. 

2:00 p.m. 

29.846 

72° 

- .117 

63° 

29.729 

2:15 

etc. 

29.839 

71 

- .114 

63 

29.725 


Aneroid Barometer Readings. 


Time. 

Place. 

Aueroid. 

Therm. 

Corr. 

Aner. 

Corr. 

Merc. 

2:10 

Office 

29".582 

74° 


29".726 

2:25 

Roof 

.518 

62 

29.663 

.723 

2:40 

Bridge 

.616 

65 

.761 

.720 

2:50 

Dock 

.632 

65 

.777 

.720 

3:05 

Office 

.576 

72 

1 - 

.722 



























57 


of the mercury to the glass, (c) Raise or lower the slide near the top 
by means of the screw on the side until a horizontal plane through 
the lower edge of the slide is just tangent to the upper surface of the 
meniscus at the top of the column, (d) Read the vernier ; the small¬ 
est reading is .002", but odd thousandths may easily be estimated if 
necessary. ( e ) To eliminate the effect of temperature on the height 
of the column, “reduce the readings to 32° F./’ by means of the 
accompanying Table I, interpolating when necessary. For exam¬ 
ple, assume readings of 29".632 and 74°; interpolating between 29.5 
and 30.0 for 74° we obtain a correction of .121, which is always sub- 
tr a elite for the range of values given in the table. 29.632 — .121 = 
29.511. (/) It is preferable to hang the mercurial barometer in a 

place where, although sheltered from the sun, it may have the same 
temperature as the external air. If this is inconvenient, a ther¬ 
mometer should be placed immediately outside the office (sheltered 
from the sun) and readings taken when the barometer is read. The 
“reduction to 32° ” should be determined from the attached ther¬ 
mometer, but the correction for temperature should be computed 
from the readings of the external thermometer and of the thermom¬ 
eter accompanying the aneroid.' (g) Plot the readings immediately 
on profile-paper (Plate A), using time as abscissae (one hour per inch) 
and reduced barometric readings as ordinates (.040" of mercury per 
inch). If the curve of pressure is very irregular, it indicates an un¬ 
stable condition of the atmosphere, or, possibly, very inaccurate 




Temp, of Ex¬ 
ternal Air. 

Approx. 
Field Read. 

Approx. 
Office Read. 

Diflf. 

Corr. for 
Temp. 

Diflf. Elev. 

63° 

308 

253 

4- 55 

+ 2 

+ 57 

62 

218 

256 

- 38 

-(+ 2) 

- 40 

60 

203 

256 

- 53 

- (+2) 

- 55 

















58 


readings of the barometer. In either case, the computed elevations 
will be very unreliable. 

Use of the Aneroid. Handle the aneroid with extreme care , be¬ 
ing especially careful that it is never violently jarred by concussion. 
Always allow it to lie flat when taking a reading. Tap it with the 
finger very gently before taking the reading. Ignore the thermom¬ 
eter inside of the aneroid and observe instead the pocket thermom¬ 
eter. If this has been carried in the pocket, allow it to remain in 
the air (shaded from the sun) long enough to indicate the true 
temperature of the air. Take readiugs* (1) beside the mercurial 
barometer ; (2) on the roof of the College Building above the Society 
rooms; (3) on the floor of the South St. bridge; (4) on the edge of 
the dock immediately underneath the bridge; (5) finally, another 
reading beside the mercurial barometer. Correct the aneroid read¬ 
ings on the roof, bridge, and dock by the mean of the discrepancies 
between the aneroid readings in the office and the reduced mercurial 
readings. Draw a curve through the ordinates of pressures and scale 
off the amounts of the ordinates for the times when the aneroid 
readings were taken on the roof, bridge, and dock. Interpolate for 
temperature at these times. With these reduced and computed 
values for the pressures and temperatures, the differences of eleva¬ 
tion may be computed from Tables II and III. For example, the 
calculation of the difference of elevation of “ Office ” and “Roof” 
is as follows: 

From Table II, for 29.663, we have 366 — (6.3 X 9.2) = 308 

Similarly, “ 29.723, “ “ 274 — (2.3 X 9.2) = 253 

Approx, diff. of elevation. 55 feet. 

62° -f- 63° = 125°. From Table III the coefficient for correction is 
.0262 + (5 X .00106) = .0315. For this and similar cases where the 
difference of elevation is small, it is only necessary to obtain from 
the table an approximate coefficient (e.g., .03 in this case) by mere 
inspection. .03 X 55 = 2, the correction, obtained only to the nearest 
foot. The correction is positive , hence the difference of elevation is 
57 feet. 


* The directions are here purely local in their application, but were introduced 
as being suggestive of methods of obtaining a total difference of elevation of 
100 feet or over, which is sufficient to illustrate the use of these instruments-. 





59 


TABLE I.—REDUCTION OF BAROMETER READING TO 32 ° F . 


Temp 

Inches . 

26.0 

26.5 

27.0 

27.5 

28.0 

28.5 

29.0 

29.5 

30.0 

30.5 

31.0 

45 

-.039 

- .039 

-.040 

-.041 

-.042 

-.042 

- .043 

-.044 

-.045 

- .045 

-.046 

46 

.041 

.042 

.043 

.043 

.044 

.045 

.046 

.046 

.047 

.048 

.049 

47 

.043 

.044 

.045 

.046 

.047 

.048 

.048 

.049 

.050 

.051 

.052 

48 

.046 

.047 

.047 

.048 

.049 

.050 

.051 

.052 

.053 

.053 

.054 

49 

.048 

.049 

.050 

.051 

.052 

.052 

.054 

.054 

.055 

.056 

.057 

50 

.050 

.051 

.052 

.053 

.054 

.055 

.056 

.057 

.058 

.059 

.060 

51 

.053 

.054 

.055 

.056 

.057 

.058 

.059 

.060 

.061 

.062 

.063 

52 

. 055 

.056 

.057 

.058 

.059 

.060 

.061 

.062 

.064 

.065 

.066 

53 

.057 

.058 

.060 

.061 

.062 

.063 

.064 

.065 

.066 

.067 

.068 

54 

.060 

.061 

.062 

.063 

.064 

.065 

.067 

.068 

.069 

.070 

.071 

55 

.062 

.063 

.064 

.065 

.066 

.068 

.069 

.070 

.071 

.073 

.074 

56 

.064 

.065 

.067 

.068 

.069 

070 

.072 

.073 

.074 

.075 

.077 

57 

.067 

.068 

.069 

.070 

.072 

.073 

.075 

.076 

.077 

.078 

.080 

58 

.069 

.070 

071 

.073 

.074 

.076 

.077 

.078 

.080 

.081 

.082 

59 

.072 

.073 

.074 

.075 

.077 

.078 

.080 

.081 

.083 

.084 

.085 

60 

.074 

.076 

.077 

.078 

.079 

.081 

.082 

.084 

.085 

.086 

.088 

61 

.076 

.077 

.079 

.080 

.082 

.083 

.085 

.086 

.088 

.089 

.091 

62 

.079 

.080 

.082 

.083 

.085 

.086 

.088 

.089 

.091 

.092 

.094 

63 

.081 

.082 

.084 

.085 

.087 

.088 

.090 

.091 

.093 

.095 

.096 

64 

.083 

.085 

.086 

.088 

.090 

.091 

.093 

.094 

.096 

.097 

.099 

65 

.086 

.087 

.089 

.090 

.092 

.093 

.095 

.097 

.099 

.100 

.102 

66 

.088 

.089 

.091 

.093 

.095 

.096 

.098 

.099 

.101 

.103 

.105 

67 

.090 

.092 

.094 

.095 

.097 

.099 

.101 

.102 

.104 

.106 

.108 

68 

.093 

.094 

.096 

.098 

.100 

.101 

.103 

.105 

.107 

.108 

.110 

69 

.095 

.097 

.099 

.100 

.102 

.104 

.106 

.107 

.110 

.111 

.113 

70 

.097 

.099 

.101 

.103 

.105 

.106 

.109 

.110 

.112 

.114 

.116 

71 

.100 

.101 

.103 

.105 

.107 

.109 

.111 

.113 

.115 

.117 

.119 

72 

.102 

.104 

.106 

.108 

.110 

.112 

.114 

.116 

.118 

.120 

.122 

73 

.104 

.106 

.108 

.110 

112 

.114 

.116 

.118 

.120 

.122 

.124 

74 

.107 

.109 

.111 

.113 

.115 

.117 

.119 

.121 

.123 

.125 

.127 

75 

' .109 

.111 

.113 

.115 

.117 

.119 

.122 

.124 

.126 

.128 

.130 

76 

.111 

.113 

.116 

.118 

.120 

.122 

.124 

.126 

.128 

.130 

.133 

77 

.114 

.116 

.118 

.120 

.122 

.124 

.127 

.129 

.131 

.133 

.136 

78 

.116 

.118 

.120 

.122 

.125 

.127 

.129 

.131 

.134 

.136 

.138 

79 

.118 

.120 

.123 

.125 

.127 

.129 

.132 

.134 

.137 

.139 

.141 

80 

.121 

.123 

.125 

.127 

.130 

.132 

.135 

.137 

.139 

.141 

.144 

81 

.123 

.125 

.128 

.130 

.132 

.134 

.137 

.139 

.142 

.144 

.147 

82 

.125 

.128 

.130 

.132 

.135 

.137 

.140 

.142 

.145 

.147 

.149 

83 

.128 

.130 

.133 

.135 

.138 

.140 

.142 

.145 

.147 

.149 

.152 

84 

.130 

.132 

'.135 

.138 

.140 

.142 

.145 

.147 

.150 

.152 

.155 

85 

.132 

.134 

.137 

.140 

.143 

.145 

.148 

.150 

.153 

.155 

.158 

86 

. 135 

.137 

.140 

.142 

.145 

.148 

.150 

.153 

.155 

.158 

.161 

87 

.137 

.139 

. J 42 

.144 

.148 

.150 

.153 

.155 

.158 

.161 

.163 

88 

.139 

.142 

.145 

.147 

.150 

.152 

.155 

.158 

.161 

.163 

.166 

89 

.142 

.144 

.147 

.150 

.153 

.155 

.158 

.161 

.164 

.166 

.169 

90 

.144 

.147 

.150 

.153 

.155 

.158 

.161 

.164 

.166 

.169 

.172 

91 

-.146 

-.149 

-.152 

- .155 

-.158 

-.160 

- .163 

-.166 

-.169 

-.172 

-.175 

































TABLE II.—BAROMETRIC ELEVATIONS.* 


B 

A 

Diff. for 
.01. 

B 

A 

Diff. for 
.01. 

B 

A 

Diff. for 
.01. 

Inches. 

20.0 

20.1 

20.2 

20.3 

20.4 

20.5 

20.6 

20.7 

20.8 

20.9 
21.0 
21.1 
21.2 

21.3 

21.4 

21.5 

21.6 

21.7 

21.8 

21.9 
22.0 
22.1 
22.2 

22.3 

22.4 

22.5 

22.6 

22.7 

22.8 

22.9 
23.0 

23.1 

23.2 

23.3 

23.4 

23.5 

23.6 

23.7 

Feet. 

11,047 

10,911 

10,776 

10,642 

10,508 

10,375 

10,242 

10,110 

9,979 

9,848 

9,718 

9,589 

9,460 

9,332 

9.204 
9,077 
8,951 
8,825 
8,700 
8,575 
8,451 
8,327 

8.204 
8.082 
7,960 
7,838 
7,717 
7,597 
7,477 
7,358 
7,239 
7,121 
7,004 
6,887' 
6,770 
6,554 
6,538 
6,423 

Feet. 

- 13.6 

13.5 

13.4 
13 4 
13.3 

13.3 

13.2 
13.1 

13.1 
13.0 
12.9 

12.9 
12.8 
12.8 

12.7 

12.6 
12.6 

12.5 

12.5 

12.4 
12.4 

12.3 

12.2 
12.2 
12.2 
12.1 
12.0 
12.0 

11.9 
11.9 

11.8 
11.7 
11.7 
11.7 

11.6 
11.6 

- 11.5 

Inches. 

23.7 

23.8 

23.9 
24.0 

24.1 

24 2 

24.3 

24.4 

24.5 

24.6 
. 24.7 

24.8 

24.9 
25.0 

25.1 

25.2 

25.3 

25.4 

25.5 

25.6 

25.7 

25.8 

25.9 
26.0 
26.1 
26.2 

26.3 

26.4 

26.5 

26.6 

26.7 

26.8 

26.9 
27.0 

27.1 

27.2 

27.3 

27.4 

Feet. 

6,423 

6,308 

6,194 

6.080 

5.967 
5,854 
5,741 
5,629 
5,518 
5,407 
5.296 
5,186 
5,077 

4.968 
4,859 
4,751 
4,643 
4,535 
4,428 
4,321 
4,215 
4,109 
4,004 
3,899 
3,794 
3,690 
3,586 
3,483 
3,380 
3,277 
3,175 
3,073 
2,972 
2,871 
2,770 
2,670 
2,570 
2,470 

Feet. 

- 11.5 
11.4 

11.4 
11.3 
11.3 

11.3 
11.2 
11.1 
11.1 
11.1 
11.0 
10.9 

10 9 
10.9 
10.8 
10.8 
10.8 
10.7 
10.7 
10.6 
10.6 

10.5 
10.5 

10.4 
10.4 
10.4 
10.3 
10.3 
10.3 
10.2 
10.2 
10.1 
10.1 
10.1 
10.0 
10.0 

- 10.0 

Inches. 

27.4 

27.5 

27.6 

27.7 

27.8 

27.9 
28.0 
28.1 
28.2 

28.3 

28.4 

28.5 

28.6 

28.7 

28.8 

28.9 

29 0 

29.1 

29.2 

29 3 

29.4 

29.5 

29 6 

29.7 

29.8 

29.9 
30.0 

30.1 

30.2 

30.3 

30.4 

30.5 

30.6 

30.7 

30.8 

30.9 
31.0 

Feet. 

’ 2,470 
2,371 
2,272 
2.173 
2,075 
1.977 
1,880 

1,783 
1,686 
1,589 
1,493 
1,397 
1.302 
1,207 
1,112 
1,018 
924 
830 
736 
643 
550 
458 
366 
274 
182 
91 

0 

- 91 
181 
271 
361 
451 
540 
629 
717 
805 

- 893 

Feet. 

- 9.9 

9.9 

9.9 

9.8 

9.8 

9.7 

9.7 

9.7 

9.7 

9.6 

9.6 

9.5 

9.5 

9.5 

9.4 

9.4 

9.4 

9.4 

9.3 

9.3 

9.2 

9 2 

9.2 

9 2 

9.1 

9.1 

9.1 

9.0 

9.0 

9.0 

9.0 

8.9 

8 9 

8 8 

8.8 

- 8.8 


* Compiled from Report of U. S. C. &. G. Survey for 1881, App. 10, Table XI. 

T1BLE III -COEFFICIENTS FOR CORRECTIONS FOR TEMPERATURE 

AND HUMIDITY.* 


t + V 

C 

Diff for 
1°. 

< + V 

C 

Diff .for 
1°. 

t + V 

C 

Diff .for 
1°. 

0° 

10 

20 

30 

40 

50 

60 

- .1024 
.0915 
.0806 
.0698 
.0592 
.0486 

- .0380 

10.9 

10.9 

10.8 

10.6 

10.6 

10.6 

60° 

70 

80 

90 

100 

110 

120 

- .0380 
.0273 
.0166 

- .0058 
+ .0049 

.0156 
+ .0262 

10.7 

10.7 

10.8 
10.7 
10.7 
10.6 

120° 

130 

140 

150 

160 

170 

180 

+ .0262 
.0368 
.0472 
.0575 
0677 
.07:9 
4- .0879 

10.6 

10.4 

10.3 

10 2 
10.2 
10.0 


> Compiled from Report of U. S. C. & G. Survey for 1881, App. 10, Tables 1, IV. 












































61 


— 23 - 

vernier CONSTRUCTION. 


Equipment: Sheet of drawing-paper 6 ' X 12", a decimal scale, a 
pair of hair-spring (or bow-spring) dividers, A hard pencil, triangles, 
etc., all of which should be furnished by the student. 

Principles involved. A vernier is a device which determines the 
position of au index-point, which is movable along the face of a scale, 
with a greater exactitude than the finest division of the scale. A 
direct vernier has n divisions which are made equal to (w—1) divisions 


on the scale. The finest reading (except by estimation) is * of the 

smallest division of the scale. On a direct vernier the vernier 
readings increase in the same direction as the scale readings. 
A retrograde vernier has n divisions which are made equal to 
(n -f- 1) divisions on the scale. The finest reading is likewise 


— of the smallest division. The vernier readings increase in a 

contrary direction to the scale readings. A double vernier, which 
may be either direct or retrograde, has a full set of vernier gradua¬ 
tions ou both sides of the index. It facilitates the use of a scale 
which is read in both directions, as is the case with the horizontal 
limb of a transit. 

Construction of scales. Construct the required scales on a single 
sheet, and cut them out afterward. Lay off the verniers by setting 
the dividers (by trial) at such a space that n repetitions of the spacing 
shall equal n — 1 (or n -f-1) divisions on the scale. 

a. Draw a scale 6 inches long, divided into fifths of an inch; also 
a single direct vernier 
by which the scale may 
be read to inch. 

b. Draw a scale 6 
inches long, divided 
into tenths of au inch; 
also a double direct ver¬ 
nier by which the scale 
may be read to inch. 

c. Draw a scale 6 inches long, divided into inches, tenths, and 
half-tenths of an inch ; also a single retrograde vernier by which the 
scale may be read to half-hundredths of an inch. 












62 

— 24 — 

SLIDE-RULE CONSTRUCTION. 

Equipment: Sheet of drawing-paper 15" X 7^", decimal scale, hard 
pencil, triangles, etc., all of which should be furnished by the 
student. 

Principles involved . An ordinary slide-rule is a combination of two 
* ‘ logarithmic scales. ” A logarithmic scale is one on which the dis- 

_ | Q SCALE B 6 C 

fo scale a a g+b a+c [ 


tance of each numbered mark from the origin is proportional to the 
logarithm of that number. Since the logarithm of the product ab is 
the sum of the lotharithms of a and of b, if the origin { o ) of scale B 
is set at the mark a on scale A, the mark b on scale B will be at a 
distance from the origin of scale A equal to the sum of the loga¬ 
rithms of a and 6, and therefore the number on scale A opposite to b 
on scale B is the numerical product of a and b. Similarly, if the 
product of a and c is desired, the relative position of the two scales 
need not be altered. It is only necessary to observe the position of 
c on scale B and note the corresponding number on scale A, which 
will be the required product of a and c. 

Since the logarithm of 1 is zero, the number at the origin is always 
1 (10, 100, 1000, etc.). The number 2 (20, 200, etc.) is always 
placed at .301030 ... of a logarithmic unit from 1. The number 8 
comes at .903090 of the logarithmic unit from the origin, and 9 and 
all the subdivisions between 8 and 10 must come between .903090 
and 1.000000. 

Since it frequently happens that when the origin of scale B is set 
at the desired point on scale A the other factor on scale B would 
come beyond the range of one logarithmic unit on scale A, it is 
necessary to give scale A a length of two logarithmic units. Two 
logarithmic units for A and one for B are all that is necessary for 
simple operations, but two logarithmic units for B are sometimes 
more convenient for the more complicated operations which are pos¬ 
sible with a slide-rule. 

Construction of scales. Lay off the scales as shown in the sketch. 
Set off between the extremities of each of the three logarithmic units 








63 


the positions for all numbers between 11 and 99 inclusive. Take the 
quantities from Table Y. ,Use the “20” decimal scale, ^V' will 
then equal .01 of the logarithmic unit. The finest space (that be- 


pro BE CUT OFF AND PASTEo"! ! PtO BE CUT OFF AND PASTEol 

I TO ONE END OF SCALE B.J || _TO ONE END OF SCALE B.J 


SCALE B 2; 

i ? ? t ? ? t ? ?rj»* y 

-r 

1 

1 

NARROW SLIT 

4 , 

“7 CD 

t O 

»- > 

^ r~ 

' m 

> 

8- 

5 £ “ 

' S 1 

! ?- 

! i-i 

L ° NARROW SLIT 

£ 

r4 

Pi- 
--i- „ 

.-.jrl -O 

ss—sriT* - v— -rr- o ; * 


L—--- A 

i 


tween 99 and 100) will then equal .0044, or slightly less than -fa". 
Estimate fractions of the smallest scale unit, ■£$”. 

Use extraordinary care in laying off the scales, marking the posi¬ 
tions with a needle-point. Ink the scales, using very fine lines. 

Cut out the narrow slits. ' Cut off scale B and lengthen it by past¬ 
ing on the two other pieces. Slide scale B through the slits so that 
it passes close to scale A. 

Test. Test the work by the solution of problems, which are veri¬ 
fied by ordinary numerical processes. When made with sufficient 
care, even such a hand-made scale will have practical value when 
used for roughly checking numerical computations whose accurate 
solution requires more tedious methods. 

Position of decimal point. Rules may be formulated for determin¬ 
ing the location of the decimal point in the final result, but their ap¬ 
plication is sometimes troublesome and liable to lead to error. A 
simpler and surer method is to make an approximate mental solu¬ 
tion. To illustrate—required the product of .37 X 680. Disregarding 
the decimal points, the significant figures are 37 and 68 and their 
product, determined by the slide-rule, has the significant figures 2516. 
An approximate mental solution shows instantly that 25.16 is too 
small and that 2516 is too large ; also that 251.6 is about what 
would be expected as the result. 















64 



— 25 — 

RAILROAD SURVEYING—SIMPLE CURVES. 

Equipment: Transit, 100-foot steel tape (or link-chain), two tran- 
sii-poles, set of marking-pins. 

Select a point from which may be run two lines making an angle - 
of about 150° and each about 300 feet long. Let this point be the 
vertex of a curve with an assumed A (e.g., 28° 46') and D (e.g., 5° 
42). 

Compute the tangent distance. Set a pin at the vertex and 
measure along the direction of one tangent a distance equal to the 
tangent distance, which will give the P.C. Set up the transit at the 
P C. and run in the curve to the P.T. Establish the direction of 
the forward tangent by a back-sight to the last transit station on the 
curve. If the vertex is visible from the P.T., sight directly to it 
and obtain the error (if any) in alignment; measure directly to it 
and compare the measurement with the computed tangent distance. 

If the P.T. and vertex are not intervisible, produce the forward 
tangent backward from the P.T. an amount equal to the tangent 


Form of Notes. 


Sta. 

Alignment. 

Vernier. 

Tang. Defl. 

Calc. 

Bearing. 

Needle. 

+02.3 

P.T. 

14° 23' 

28° 46' 

N 47° 10' W 

N 47° 15' W 

15 

£ 88 

14° 15' 




14 


11° 24' 




13 

3 § 5 « 

8° 32' 




12 

- o 'S 

a 

5° 42' 




11 

& H 

5° 51' 




10 

P.C. 



N 18° 24' W 

N 18° 60' W 














65 


distance, aud observe the discrepancy between this point and the 
vertex. 

The value of this exercise is increased when the whole curve is 
not visible from the P.C., thereby rendering necessary one or more 
transit-stalious along the curve. In such cases, and especially when 
the degree of the curve is an odd quantity, the best plan is to com¬ 
pute all deflections as if the whole curve were visible from the P. C., 
checking the computation for the last substation by observing its 
equality to £ a. With the transit at any intermediate station along 
the curve, instead of following the usual rule and having the 
verniers at zero when the telescope is tangent to the curve, point at 
the previous tiausit station with the verniers reading the deflection 
for that station. Plunge the telescope aud point at any forward 
station with the same deflection as was computed for that station for 
the transit at P.C. In this way the original list of deflections can 
be followed and possible error avoided. For example (see below), 
with the transit at sta. 14, sight bp,ck to sta. 12 with a reading of 5° 
42'. When the telescope is plunged and the verniers set at 11° 24' 
the telescope will evidently be tangent to the curve. An additional 
deflection of 2° 51' is necessary to point to sta. 15, and 14° 15' is the 
computed deflection for sta. 15. The deflection for any station, 
visible or not, is the same as from the P.C. 


(Read up the page.) 


Error of alignment, 0° 01)4' 
“ “ distance, O'.2 







66 


— 26 — 

RAILROAD SURVEYING—SIMPLE CURVES, VERTEX 
INACCESSIBLE. 

Equipment: Transit, 100-foot steel tape (or link-chain), two transit- 
poles, set of marking-pins. 

Assume and locate the two tangents by setting two marking-pins 
for each, so that the central angle will be about 30°, and so that the 
two tangents would intersect within some building or other inacces¬ 
sible place. Assume D at some value (e.g., 5° 26') so that the 

curve will be about 500 to 600 
feet long. Suppose a, b, c, and d 
are the marking-pins. Set up 
the transit at b ; take a back¬ 
sight to a and then point to c, 
thus measuring the angle vbc. 
Set up at c and similarly obtain 
the angle bcv. Measure be, and 
theu compute by trigonometry vb and vc. vbc-\-bcv = A. From A 
and D the tangent distance may be found, and, knowing both this 
and vb, the P.C. may be located from b. Run in the curve to P.T. 
Check alignment by the direction of cd, and distance by measuring 
from c to P.T. 

Notes. —Similar to those for Prob. 25. Make a sketch (similar to 
that above) on the right-hand page, recording all the necessary 
computations. 





67 


- 27 - 

RAILROAD SURVEYING—TRANSITION CURVES. 

(Searles’ Railroad Spiral.) 

Equipment: Transit, 100-foot steel tape (or link-chain), two transit- 
poles, set of marking-pins. 

Select a suitable point as the vertex of a curve about 600 feet long, 
having a central angle of about 30°. Assume definite values for A 
and D and select a suitable spiral. Compute the tangent distance 
( T s ) for the curve and spiral. Lay off this tangent distance from the 
vertex ; set up the transit at this point and run in the spiral. Set up 
the transit at the other end of the spiral and verify the setting of all 
the spiral points by back-sights. Run in the circular curve, lo¬ 
cating its middle point (opposite the vertex), to the beginning of the 
terminal spiral. Set up the transit there and locate that spiral to the 
P.T. Finally set up at the P.T., verify the points of the terminal 
spiral, and check the location of the P.T. similarly to Prob. 25. 
Measure the distance from the middle point of the curve to the ver¬ 
tex and compare the measurement with the computed external dis¬ 
tance ( E s ). 

Notes.— Same form as for Prob. 25. The chord points of the 
spiral are recorded as substations with the proper deflection for each 
in the “vernier” column. The elements of the spiral chosen are 
recorded in the “alignment” column similarly to those for circular 
curves. 


G8 


— 28 — 

RAILROAD SURVEYING—CROSS-SECTIONING. 

Equipment: 50-foot tape, hand-level, 5-foot rod, level-rod (or light 
stadia-rod). Also, set of cross-section rods.* 

1. With hand-level. Cross-section for 100 feet on each side of a 
line of stakes, already set by a previous survey. Obtain distance and 
elevation for every “ break” in the slope, representing distance as the 
denominator of a fraction and elevation as the numerator. Eleva¬ 
tion and distance are always referred to the centre. 

2. With cross-section rods. Repeat the cross-sectioning, deter¬ 
mining distance and elevation at every 10-foot point as well as at all 
breaks. 

3. Plot one set of cross-sections on cross-section paper (10 lines to 


Form of Notes. 


Sta. 

Surf. 

Elev. 

Grade 

Elev. 

Center 
cut (+) 
or fill (-). 

Grade. 

etc. 

+ 60 

etc. 

26.0 

etc. 

36.2 

etc. 

-10.2 


28 

26.2 

35.8 

- 9.6 


27 

38.4 

25.2 

+ 3.2 


26 

42.7 

34.6 

+ 8.1 


25 

44.6 

34.0 

+10.6 

+ 0 . 6 % 


* The cross-section rods referred to are a set of three rods, two of them 8 
feet long, divided into feet and tenths, and used vertically. The third rod is 10 
feet long, divided also into feet and tenths and provided with level-bubbles set 














69 


the inch, each way), using the scale of 10 feet to the inch, both for 
horizontal and vertical scales. The “ elevations” (second column of 
Notes), which are obtained from a previous levelling, are only used 
in the plotting. Using the heavy lines on the cross-section paper as 
even 10-foot elevations, plot first the elevation of “ centre,” and then, 
by addition or subtraction, the elevation of each point taken. A line 
drawn through these points will represent a cross-section of the sur¬ 
face. The intersection of this line by the 5 (10 or 20) foot lines of 
elevation gives graphically the distance of the 5 (10 or 20) foot con¬ 
tours from the centre. The “ grade elevations ” and “ grade ” may 
be assumed (for the sake of this problem) if they have not been 
definitely determined, and the “centre cut or fill” computed. As¬ 
sume the width of road-bed as 20 feet in cut and 14 feet in fill. 
Assume slope ratio of 1£ horizontal to 1 vertical. Plot the outline 
of the cut (or embankment) on each cross-section. 

4. Plot the centre line and contours on drawing-paper 


(Read up the page.) 


Left. 


Right. 


etc. 

- 1 - 14.6 + 11.7 + 9.8 + 2.6 + 2.1 


100 “ 60 29.0 13.6 9.4 

etc. 


etc. 

- 3.3 - 6.6 - 9.7 - 11.7 
' 45 ^ 0 ' 62.0 100 


etc. 



method of use is obvious. 






















70 


- 29 - 

railroad SURVEYING^SETTING SLOPE-STAKES. 

Equipment: Level, level-rod (or light stadia-rod), 50-foot tape. 

Location : Usually this work is done on a line of center stakes 
whose elevation above datum and above or below “grade” is 
known. The columns of the left-hand page (see below) should be 
filled out before commencing work. The same essential practice 
may be obtained by selecting a point on a side hill as “center” 
-and then successively ascribing various depths of cut or fill to it, 
each new depth corresponding to different positions of slope-stakes. 



Method a. Determine the location of the slope-stakes from the 
formula x — \b + sd ± sy, in which s is the slope ratio. First, com¬ 
pute at any station b -{- sd —giving the point a. Then estimate by 
eye how much further than this a point must be so that the added 
distance will be 1^ times the added difference of elevation. Set up 
the rod at this total estimated distance. If the difference of eleva¬ 
tion (y) satisfies the above formula, the required point has been 
found. If not, a little study will show whether the point should be 


Form of Notes. 


Sta. 

vSurf. 

Elev. 

Grade 

Elev. 

Center 
cut (+) 
or fill (-) 

Grade. 

24 + 60 

90.8 

99.2 

-8.4 


24 

92.4 

99.8 

- 7.4 

- 1.0# 






















71 


nearer to or further from the center and about how much. With 
practice, two trials will generally he sufficient to locate a point with 
sufficient accuracy. The upper side of the fill is determined simi¬ 
larly, except that the correction ( sy ) is negative. 

An easy check on the numerical calculation is found by subtract¬ 
ing \b from the computed distance and noting whether the difference 
is times the difference of elevation. For instance, take the result 


-18.6 


36.9 — 9.0= 27.9, which is 1£ times 18.6; again, 15.3 — 9.0 


36.9' 

= 6.3, which is 1£ times 4.2. 

/ — 14.2\ 

Determine the distance and elevation of all “ breaks” ^e.g. -— 20 ^ 5 / 


between the center and the slope-stakes. 

Method b. Using an automatic slope-rod and a specially marked 
distance-tape. 

The rod is ten feet long and has an endless graduated tape 20 feet long rolling 
over friction wheels at the ends, which are shod with metal shoes. The distance- 
tape is graduated on one side as usual; on the other side a zero-point is marked 
9 feet (or more) from the ring. From the zero back to the ring the tape is 
graduated to feet and half-feet. Beyond the zero-point the tape is graduated 
to a scale s times the usual scale. S is usually 1}. 

(1) Set the rod-tape at zero—i.e. so that the zero is at the bottom 
of the rod. (2) Hold the rod at the center stake (c) and note the 

reading k. (S) If * is | ^ s e s ater | tlian d, j the tape on the 

back side of the rod by an amount equal to j ^ ~ ^ j-. (4) With the 

distance-tape, so held that its zero is \b from the center, carry the 
rod out until the rod reading equals the reading indicated by the 
tape. The rod will then be at the required point. The proof is 
apparent from the figure. 


[Read up the page.] 


Left. 

T> . . . 5=18' in fill, 23' in cut. 

Right, s = H : 

-18.6 


- 14.2 

- 2.2 

- 4.2 

36.9 


20.5 

8.5 

15.3 

-16.2 


- 5.4 

- 1.6 

- 3.4 

33.8 


18.6 

10.0 

14.1 

















72 


— 30 - 

railroad SURVEYING—TURNOUT, FROM EXISTING 

TRACK. 

Equipment: Transit; 100-foot steel tape; 50-foot tape divided to 
feet, tenths, and hundredths; set of marking-pins; and a Railroad 
Hand-book with Tables. 

Location: It is usually possible to find a piece of constructed 
curved track where this problem may be worked. A convenient 
“ siding” that is but little used will generally be a suitable place, 
since fchere will then be but little danger of interference from trains. 

Radius of the curved track. The radius is best determined by setting 
up the transit in the centre of the track (preferably at the desired po¬ 
sition for the heel of the stub-switch or the head-block of the point- 
switch) and measuring off 100 feet in each direction to points which 
are also in the centre of the track. Sight the transit at one of these 
points, with the plates reading 0°; plunge the telescope and sight at 
the other point. The angle turned off equals the degree of the 
curve.* 

Preliminary computations. Select the frog number and compute 
the radius of the switch rails, the distance (BF) from the heel of the 
switch to the frog; also the distance ( af) from the centre of the track 
opposite the heel of the switch to the centre of the switch-track op¬ 
posite the frog. Also compute the radius and central angle of the 
“ connecting curve ” required to connect the switch-rails with a track 
parallel to the main track. (See Searles’ Field Engineering, §§ 186, 
187, and 194 ; or Webb’s Railroad Construction, §§ 267, 268, 272, 273. 

Field-work. 1. Locate the frog-point by a measurement (BF) from 
the position selected for the heel of the switch. Set up the transit at 
the centre of the track at the heel of the switch (a). Set off the di¬ 
rection of the tangent by turning off \ the degree of curve of the 
main track from a point in the centre of the main track 100 feet 
away. Set off the switch-track curve by 20-foot chords. If the 
curvature of the switch-rails is sharp, the nominal 20-foot chords 

* When no instrument is at hand, the radius may be found by stretching a 
tape or even a string between two points on the inner side of the head of the 
outer rail and measuring the middle ordinate (m). Measure the chord (c). The 
radius = c a 8m. Although this formula is not theoretically exact, the error 
is extremely small and is always far within the accuracy obtainable when 
using a string or tape. Of, course this gives the radius of the outer rail and 
i-gauge should be subtracted to obtain the radius of the centre of the track. 




73 


should be increased slightly (see Searles’ Field Eng., § 107). Check 
the work by noting whether the point (/) in the centre of the switch- 
track opposite the frog, as located by deflection and distance, comes 
at ^-gauge from the frog-point ( F ). 

2 . Set up the transit in the centre of the switch-track opposite the 
frog-point (/). Determine the tangent by a back-sight to the previous 
transit-point and locate the “connecting curve.” Check by noting 
whether the last point of the connecting curve is at the proper dis¬ 
tance ( p ) from the centre of the main track. 

Notes. —The report should consist of a concise statement of the 
work done, with all the computations in full arranged in an orderly 
manner—as in the specimen solution given in §186 of Searles’ Field 
Engineering. 

Equations to be solved : 



g — gauge ; n — frog number; p = distance between centres of 
parallel tracks; F= frog-angle; B = Oa ; / = Ca; r" = Qf. 

on 

tan ^0 = ; 


BE = 2(R -f £#) sin |0; 
of = 2 r' sin UF — 0); 

tan ^ = aK 2n: 

= + + 





74 


- 31 - 

level-trier, 

Equipment: Level trier; level-tube to be tested, which may be an 
unattached tube or a tube in an instrument which may be placed 
bodily on the level-trier. 

Location : For accurate work the level-trier must be placed where 
it is perfectly steady and not subject to vibrations; as, for instance, 
on a shelf fastened to a masonry wall, or on a solid masonry pier. 

Constants of the trier. It is necessary to know the pitch of the 
screw and the length of the arm. The determination of these, with 
such aecuracy as is neeessary, requires that the screw be taken com¬ 
pletely out, involving excessive wear if done frequently and by inex¬ 
perienced hands. These values should therefore be determined 
definitely for the particular instrument used by the instructor in 
charge and the values given to the student. 

Value of one division of the level-tube. Place the level-tube (or in¬ 
strument) on the trier. If the large telescope level-tube of a transit 
is to be tested, place the whole instrument (including the levelling- 
head) on the flat plate of the trier provided for that purpose. Level 
up the transit by means of its levelling-screws, being very particular 
to orient the instrument so that the tube to be tested is exactly paral¬ 
lel to the axis of the trier. Use the transit levelling-screws to set the 
bubble at any desired position in the tube and thus save wear on the 
fine micrometer-screw. Bring the bubble near one end of the tube, 
yet so that the micrometer-index points to some even tenth or twen¬ 
tieth division or preferably at zero. Note the position on the scale of 
both ends of the bubble—estimating tenths of a division. A micro¬ 
scope will be of assistance in estimating tenths. Turn the micrometer 
to the next even fifth (tenth, or twentieth) division. Again read and 
record the position of the ends of the bubble. Continue these 
operations until the bubble is near the other end of the tube. The 
bubble always drags somewhat behind its true position; therefore, 
when moving the bubble through the tube, the screw should always 
be turned in the same direction and never reversed. When the bub¬ 
ble has been moved as far in one direction as desired, turn the screw 
a few divisions farther in that direction and then turn backward to 
the same micrometer reading it had before. On account of lagging, 
the bubble readings will be somewhat different. Move the screw 


75 


backward to each reading in turn as before. The bubble readings 
will differ from those before obtained for the same micrometer read¬ 
ings, and the differences will be practically constant and will equal 
twice the effect of the lagging. The position of the centre of the 
bubble is found by taking one-half of the difference of the two end 
readings, which gives the distance of the centre of the bubble from 
the centre of the tube. The difference in the positions of the centre 
of the bubble for corresponding settings of the micrometer-screw 
gives twice the effect of the lagging. This quantity should be 
determined in order to obtain a measure of the uncertainty due to 
this cause in any random reading. The motion of the centre of the 
bubble for two consecutive positions of the bubble is a measure of 
the length of bubble-tube corresponding to a definite angular motion. 
Since the micrometer intervals are equal, the average difference is 
the mean value corresponding to that micrometer interval. The sum 
of the bubble readings at each end is a measure of the length of the 
bubble. The differences in the bubble lengths may be caused by (a) 
variation in temperature, (b) variation in calibre of tube, (c) errors of 
reading. If the temperature is fairly constant and the tube presu¬ 
mably of good workmanship, any large variation will probably be due 
to the third cause. When the bubble is sluggish or excessively 
sensitive, considerable time (three to five minutes) may be required 
before the bubble becomes stationary. 

If a 1 ' = angular value in seconds of arc of one division of the 
index-head of the screw, 

n = number of micrometer divisions turned between each 
position of bubble, 

d = mean movement in divisions of the bubble, 

TtOC 1 r 

then angular value of one division of bubble = —— = D. 

a 

Radius of the tube. Let m — number of divisions per inch on the bub- 

TTiTlCX. 1 ' 

ble-scale ; then ——— (= mD) is the angular value of one inch of the 

CL 

tube. Since there are 206 265 seconds in the arc which equals its 


radius, the radius of the tube in inches = 







Form of Notes. 


No. 

Disk 

Bubble Readings. 

Position 

of 

Centre 

of 

Bubble. 

Motion 

for 

Consecu¬ 

tive 

Positions. 

Change of 
Position- 
Same 
Screw 
Reading. 

Reading. 

Left. 

Right. 

X 

2 

40 

60 

6.1 

7.9 

23.8 

22.0 

- 8.85 

- 7.05 

1.8 

2.05 


3 

80 

10.0 

20.0 

- 5.00 

1 55 


4 

100 

11.5 

18.4 

- 3.45 

1+0 

1.70 

1 65 


5 

120 

13.3 

16.8 

- 1.75 


6 

140 

15.0 

15.1 

- 0.05 


7 

160 

16.8 

13.6 

+ i: 6 

i iso 

1.5 

1.7 

2.3 


8 

180 

18.6 

11.8 

4- 3.4 


9 

200 

20.0 

10.2 

+ 4.9 


10 

220 

21.8 

8.6 

+ 6,6 


11 

240 

24.1 

6.3 

+ 8.9 


11 

240 

24.6 

5.9 

+ 9.35 

2.15 

1.95 

1.75 

1.5 

1.7 

1.65 

1.85 

1.75 

1.7 

1.75 

+ 0.45 

10 

220 

22.4 

8.0 

+ 7.2 

.60 

9 

200 

20.5 

10.0 

+ 5.25 

' .35 

8 

180 

18.7 

11.7 

+ 3.5 

.10 

7 

160 

17.2 

13.2 

+ 2.0 

.40 

6 

140 

15.5 

14.9 

+ 0.3 

.35 

5 

120 

13.9 

16.6 

- 1.35 

.40 

4 

100 

12.1 

18.3 

- 3.1 

.35 

3 

•80 

10.2 

20.1 

- 4.95 

.05 

2 

60 

8.6 

21.9 

— 6.65 

.40 

1 

40 

6.9 

23.7 

- 8.4 

[ +0.45 


1.775 0.354 


The “ plate-bubbles ” on a transit are too coarse to require such 
refined methods of work. It is advisable, however, to obtain some 
idea of their sensitiveness. This may be done by observing how 
much motion of the micrometer-screw is required to move the bubble 
some definite amount—say 5 y'. The screw motion, reduced to 
angular measure, gives the angular value of such a motion of the 
bubble, and therefore of such an error in the adjustment of the 
bubble. 


























77 


Length 

of 

Bubble. 


29.9 

29.9 

30.0 

29.9 

30.1 

30.1 

30.4 

80.4 

30.2 

30.4 
30.4 


30.5 

30.4 

30.5 
30.4 
30.4 

30.4 

30.5 

30.4 
30.3 

30.5 
30.0 


Length of arm—17.92 inches 
Pitch of screw—60 per inch 
One div. of disk raises arm x = BO V 0 inch 
a : 206 265:: : 17.92 

a = 1".918 


n = 20; d = 1.775 div. 

d 1.775 

m = 15 div. per inch 

Rad. of tube in inches = 


206 265 


= i5ir2a = 687 

637 inches = 53.1 feet 

Mean lagging of bubble = — = 0.177 div. 


0.177 div. = 3".8. 


When testing a transit, test both plate-bubbles as just given, and 
also test the large telescope-bubble by the accurate method. 

The uniform interval for the screw readings should be varied 
according to the sensitiveness of the bubble; a very sensitive bubble 
may move about-one bubble division for one or two divisions of the 
screw, while a coarser bubble may require a movement of 5,10, or 20 
divisions of the screw to move the bubble one division. 









78 


— 32 — " 

INEQUALITY OF TELESCOPE COLLARS OR OF 
TELESCOPE AXIS PIVOTS. 

Equipment : Telescope of a wye-level or the telescope of a theod¬ 
olite having open wyes, and a suitable striding-level. When testing 
the collars of a theodolite telescope, dummy wyes may be provided, 
which will leave the horizontal plates free and available for other 
tests—such as those for eccentricity, errors of runs, etc. 

Location : [See Problem 31.] 

Method: Set the telescope in its wyes, place the striding-level on 
the pivots (or collars), and level up until the bubble is nearly in the 
centre. Record the readings of both ends of the bubble. Reverse 
the striding-level and record the bubble readings. Reverse the 
telescope in the wyes and apply the striding-level both in direct and 
reversed position. Again reverse the telescope in the wyes and 
proceed as before until three sets of observations have been taken 
with the telescope in both positions in each set. From the figures 
obtained ten values of the inequality of the collars may be deduced. 
Compute the mean value and the probable error of a single observa¬ 
tion. 

Form of Notes. 


Tele¬ 

scope. 

Striding- 

level. 

Bubble. 

Centre of 
bubble.- 

Motion of 
bubble for 
alt. readings. 

d 

d a 


i Direct 

22.7 j 

22.3 

+ 0.2 

i 



Direct 

1 Reversed 

24.5 j 

20.3 

+ 2.1 

V . ....4.90 

.395 

.156025 


j Direct 

27.3 j 

17.1 

+ 5.1 

] L..5.25 

.045 

.002025 

Inverted 

IReversed 

29.5 | 

14.8 

+ 7.35 

[■ J ... 6.05 

.755 

.570025 


j Direct 

21.0 j 

22.9 

- 0.95 

V....5.45 

.155 

.024025 

Direct 

i Reversed 

23.8 ! 

20.0 

+ 1.9 

> \ ....5.25 

.045 

.002025 


j Direct 

26.0 j 

17.4 

+ 4.3 

v....4.85 

.445 

.198025 

Inverted 

1 Reversed 

28.4 j 

14.9 

+ 6.75 

> | ....5.75 

.455 

.207025 


j Direct 

20.1 | 

23.0 

- 1.45 

>_5.05 

.245 

.060025 

Direct 

I Reversed 

23.2 j 

19.8 

+ 1.7 

t ....5.30 

.005 

.000025 


j Direct 

25.8 | 

18.1 

+ 3.85 

' V ...5.10 

.145 

.038025 

Inverted 

i 1 Reversed 

28.8 | 

15.2 

+ 6.8 

) 





i 



52.95 


1.257250 


i 

1 




5.295 

1 


















79 


Principles involved. The position of the centre of the bubble can 
always be determined by taking one-balf of the difference of the 
readings of the bubble ends, and this is true 
regardless of whether the graduations actually 
begin in the centre or at some distance each 
side of the centre. The movement of a bub¬ 
ble is then found by determining the motion 
of the centre , and its angular value by multi¬ 
plying the movement in divisions by the an¬ 
gular value of one division. If a represents 
the diameter of one collar and b that of the 
other (the inequality is grossly exaggerated in the figure), then 
b — a is the difference of diameter. If we consider the telescope 
changed in the wyes but the level unchanged (relative to the wyes), 
the level will be inclined by the angle a, which is the angle sub¬ 
tended by 2 (b — a) at a distance of l. This is four times the angle 
between the axis of the collars and the line joining the bases of the 
striding-level, which is the required angle; i.e., /3, the required angle, 
is equal to la. Therefore the required angle is one-fourth of the 
angular value corresponding to the motion of the centre of the bubble 



motion of 
bubble for 
reversal. 


0.950 


! 

I 

! 

1 

) 

r 


1.125 

1,425 

1.225 

1.575 

1.475 

7.775 

1,296 


d 

d 2 


.346 

.119716 

E x = .6745aA 2 ;J 725 = .252 

.171 

.029241 

j? o = -^U.080 

VlO 

.129 

.016641 

„ 5.295 .080 

Error due to collars = —— ± —— 

4 4 

.071 

.005041 

= 1.324 ± .020 

.279 

.077841 

JV = .6745 V -f- = .160 

.179 

.032041 

E 0 ' = — = .065 

V6 


,280521 

Error of level adj. = 1.296 ± .065 


The right-hand pivot (telescope direct) is too large. The right leg of the 
striding-level (when direct) is too high. 






























80 


due to a change of the telescope in the wyes but no change of the 
striding-level relative to the wyes. This- is found by comparing 
alternate readings of the striding-level when used as described 
above (“Method”). 

Error of adjustment of the bubble-tube. As this is determinable 
from the observations just described, without additional instrumental 
work, the method of computation is here given. One-half of the 
algebraic sum of the distances of the centre of the bubble from the 
ceutre of the tube, for readings of the bubble when the level is 
reversed but the telescope is unchanged in the wyes (readings 
toward one end of the tube being considered positive and toward the 
other end negative), gives the value in bubble divisions of the error 
of adjustment of the bubble-tube. The above-described observations 
will give six values of this error. Compute the probable error of one 
determination. 


— 33 — 

THEODOLITE WITH MICROMETERS — EXERCISES AND 

TESTS. 

Equipment ; Only the lower part of the instrument is needed for 
this test. The telescope, striding-level, etc., may be removed and 
used in other problems if desired. The instrument may be set up 
in any convenient place where a good light may be obtained for 
illuminating the micrometers. The instrument need not be levelled. 
If the light comes from ouly one direction (a window), it is best to 
keep the lower plate loose so .that it may be turned on the levelling- 
screws and allow the micrometers to always have the same illu¬ 
mination. 

Method: Set the micrometers so that the centre of the comb of 
micrometer “A” is approximately at 0° 2' -f ; the centre of the 
comb of micrometer “B ” (assuming that the eccentricity is not ex¬ 
cessive) will read 180° 2' + . Disregard the degrees of micrometer 
B, only noting the minutes and seconds. Turn the micrometer- 
screw of micrometer A until the movable parallel wires are directly 
over (or equidistant from) the 0° mark. Note from the comb the 
minutes and from the wheel the seconds. Turn the screw so as to 
bring the parallel wires equidistant from the 0° 05' mark. When 
the micrometer is properly adjusted, the reading on the wheel in 


.81 


this position should nearly agree with the reading when the wires 
were over the 0° mark. Obtain two readings similarly with microm¬ 
eter B. Record the readings of seconds on the micrometer-wheels, 
enclosing corresponding readiugs in braces (as shown on p. 56). It 
is desirable (especially for a novice) that after a reading of the micro¬ 
meters is taken the parallel wires should be moved out of position 
slightly and again set and read. If this is done several times, vari¬ 
ous values, having a range of perhaps several seconds, will probably 
be found. The extreme range of several such readings will give an 
idea of the closeness obtainable. In such cases a mean value should 
be chosen. Turn the upper part of the instrument until micrometer 
A is at 20° 2' + and clamp it. Then turn the whole instrument until 
the micrometers are substantially in the same position as before and 
have the same illumination. Again read micrometers A and B as 
before. Repeat this operation for the 18 different positions around 
the circumference. 

Subtracting the first micrometer reading from the second gives 
the “ error of runs.” This should be substantially the same for all 
positions and should average not more that 5" and as much less as 
is possible. 

Find the eccentricity of the micrometers and the eccentricity of 
the centre of the vertical axis. (See Appendix C, p. 96 et seq.) 

[See pp. 56 and 57 for Form of Notes.] 


82 


Form of Notes. 


Micr. A. 

Micr. 

B. 

Error of Runs. 

B-A 

B—A— v 

2D 

D 

A 

B 

Of // 

/ 

// 

// 

// 

// 

If 



/25 

/ 

21 







0 02 \26 

02 V 

22 

+ 1 

+ 1 

- 4.0 

- 3-6 

- 6.5 

- 3.2 

/ 26 

/ 

17 





• 


20 02 \26 

02 V 

20 

0 

+ 3 

- 7.5 

- 7.1 

- 8.0 

- 4.0 

/31 

/ 

22 







- 40 02 \30 

02 V 

28 

- 1 

+ 6 

- 5.5 

- 5.1 

- 4.0 

- 2.0 

/18 

/ 

10 







60 02 \18 

02 ( 

14 

0 

+ 4 

- 6.0 

- 5.6 

- 9.0 

- 4.5 

/38 

/ 

34 







80 02 \ 37 

02 V 

38 

- 1 / 

+ 4 

- 1.5 

- 1.1 

- 3.5 

- 1.7 

/ 36 

/ 

36 







100 02 V36 

02 ( 

.40 

0 

+ 4 

+ 2.0 

+ 2.4 

- 1.0 

- 0.5 

/ 32 

/ 

'37 







120 02 \30 

02 \ 

.40 

- 2 

+ 3 

+ 7.5 

+ 7.9 

+ 6.5 

+ 3.2 

/ 36 

/ 

'33 







140 02 V35 

02 V 

35 

- 1 

+ 2 

- 1.5 

— 1.1 

+ 1.0 

+ 0.5 

/ 37 

/ 

'42 







160 02 \42 

02 V 

.40 

+ 5 

- 2 

+ 1.5 

+ 1.9 

+ 2.5 

+ 1.2 

/ 25 

/ 

'28 







180 02 \22 

02 V 

.24 

- 3 

- 4 

+ 2.5 

+ 2.9 



/ 36 

/ 

'36 







200 02 \34 

02 ( 

35 

- 2 

- 1 

+ 0.5 

+ 0.9 



/ 29 

/ 

'26 







220 02 \29 

02 V 

29 

0 

+ 3 

- 1.5 

- 1.1 



/19 


'21 







240 02 Vl7 

02 ( 

21 

- 2 

0 

+ 3.0 

+ 3.4 



/ 31 


'34 







260 02 \31 

02 ( 

32 

0 

- 2 

+ 2.0 

+ 2.4 



/19 


'21 







280 02 \21 

02 ( 

25 

+ 2 

+ 4 

+ 3.0 

+ 3.4 



/24 


'24 







300 02 \24 

02 l 

.26 

0 

+ 2 

+ 1.0 

+ 1-4 



/17 

/ 

'14 







320 02 \21 

02 V 

.19 

+ 4 

+ 5 

- 2.5 

- 2.1 



/ 25 

/ 

'22 







340 02 V^21 

02 V 

.22 

- 4 

0 

- 1.0 

- 0.6 






+12 

+41 

+ 23.0 







-16 

- 9 

- 31.0 







- 4 

+32 

- 8.0 







- 0.2 

+ 1.8 

v — — 0.4 
































83 


sin t 

cos t 

D sin t 

D cos t 

t-,k 

2e" sir ( t-h ) 

g 





O 

ft 

n 

0.00 

1.00 

0 

- 3.20 

- 107 

- 3.7 

+ 0.1 

.34 

.94 

- 1.36 

- 3.76 

- 87 

- 3.9 

- 3.1 

.64 

.77 

- 1.28 

- 1.54 

- 67 

- 3.6 

- 1.5 

.87 

50 

- 3.91 

- 2.25 

- 47 

- 2.8 

- 2.e 

.98 

17 

- 1.67 

- 2.89 

- 27 

- 1.7 

+ 0.6 

.98 

- .17 

- 0.49 

+ 0.08 

- 7 

- 0.5 

+ 29 

.87 

- .50 

-f 2.78 

- 1.60 

+ 13 

+ 0.9 

+ 7.0 

.64 

- .77 

4-0.32 

- 0.38 

+ 33 

+ 2.1 

- 3.2 

0.34 

- .94 

4-0.41 

- 1.13 

+ 53 

4 3.1 

- 1.2 





+ 73 

4. 3.7 

-0.8 


- 



+ 93 

+ 3.9 

- 3.0 





+ 113 

+ 3.6 

- 4.7 





+ 183 

+ 2.8 

+ 0.6 





4 153 

+ 1.7 

+ 0.7 





4 173 

+ 0.5 

+ 3.9 





+ 193 

- 0.9 

+ 2.3 





+ 213 

- 2.1 

0.0 





+ 233 

- 3.1 

+ 2.5 



- 8.71 
+ 3.51 

- 16.75 
+ 0.08 



+ 20.6 
- 20.4 



- 5.20 

- 16.67 

1 




S (D cos J.) = _ tan h _ ~ 16 -g? = 3 . 2 O; tan h = - 3.20; /i = 107° 21' (call it 107°). 
2 (D sin t) ~ 5.20 

- 2 {D cos t) _ + 16,67 j „ g g . the centre is toward the 107° mark. 
e ~ 9 sin 107° 21' + 8.64 

R = 4.5 inches; e = 1.93 X 4.5 X .0000048 = .000042 of an inch. 

Note that the mean value of g is almost exactly zero, as the theory requires. 






























84 


— 34 — 

MODULUS OF ELASTICITY OF STEEL TAPE. 

1 

Equipment: 300' (or 100') steel tape, provided at one end with a 
very fine scale (preferably with a vernier) so that very minute 
changes of length may be detected ; a spring balance or other appa¬ 
ratus for measuring tension, which should be determined with great 
accuracy ; if a screw and nut is so arranged at each end that the tape 
may be held at any required tension without any unsteadiness, the 
accuracy will be greatly increased. 

Location: These tests can best be made between fixed hubs or 
monuments at some place which is permanently arranged for them. 
The space under the seats of a grand stand will often prove a favor¬ 
able place for such work. 

Method: If it is readily possible to support the tape throughout 
its length, the effect of sag may be eliminated ; otherwise the tape 
must be supported by freely swinging hooks set at regular intervals. 
Set the tape in place so that it swings freely in the hooks, with the 
zero at one end exactly on the index-mark on the monument at that 
end. Stretch the tape by means of the screws until the pull is 12 
lbs. and take the exact reading on the fine scale at the other end of 
the tape. Slacken up the tape slightly and again stretch it to 12 lbs. 
and again take the reading. Take three such readings with tensions 
of 12, 14, 16, 18, and 20 lbs. 


Form of Notes. 


Tension. 

Vernier Scale. 

Mean. 

P 2 &P, 

u a —V, 

c a 


1 0.0292 





12 lbs. 

■< 0.0291 

0.02913 

16, 12 

0.02087 

0.00128 


(0.0291 






(0.0395 


18, 14 

0.02057 

0.00108 

14 lbs. 

\ 0.0393 

0.03940 





(0.0394 


20, 16 

0.02007 

0.00075 


(0.0500 





16 lbs. 

■< 0.0500 

0.05000 

18, 12 

0.03084 

0.00206 


(0.0500 






(0.0599 


20, 14 

0.03067 

0.00139 

18 lbs. 

< 0.0600 

0.05997 





(0.0600 


20, 12 

0.04094 

0.00237 


(0.0700 





20 lbs. 

•< 0.0701 

0.07007 





(0.0701 

























85 


Computations : Since the tape was swung on hooks spaced 30 feet 
apart, there is a difference in the sag for the various tensions, which 

equals C 8 = in which L = length of tape in 

feet (300), to = weight* of one linear unit of the tape in pounds 
(0.000574), d = the uniform distance between supports in the same 
unit as w, and Pi and are the various tensions. If h and h are the 
true lengths, Vi and v 2 the mean vernier readings, and s x and the 
sags for two tensions, h = v l -\- $i and U = v* -f-Sa. The stretch a 


* It is difficult to obtain with accuracy the weight of a linear unit and also the 
cross-section. The weight of a linear unit is sometimes obtained by weighing 
the whole tape and dividing by the length, and then the cross-section is ob¬ 
tained by dividing the weight of a linear unit by the assumed weight of a cubic 
unit of steel of that quality. These results are vitiated by the fact that handles, 
rivets, brass sleeves, etc., which are not readily removable add to the weight of 
the tape and also because the weight of a cubic unit is rather uncertain. To ob¬ 
tain the cross-section accurately by direct measurement is impracticable, be¬ 
cause the cross-section is not truly rectangular, the sides of the cross-section 
being more nearly circular, and again because the-thickness (generally about 
.015") cannot be readily measured, even with micrometer calipers, with a suffi¬ 
cient percentage of accuracy. When it is possible to obtain from the manufac¬ 
turer a sample of the same kind of tape, the following method is more accu¬ 
rate 1 A piece of the tape 2.52 inches long was weighed on a chemical balance 
as 656.2 m.g. (= .0014467 lb). It was also weighed in distilled water having a 
temperature of 16°.4 C., the weight being 572.4 m.g. From these figures and the 
known weight of distilled water the weight of the steel per cubic inch was de¬ 
termined to be 0.2825 lb. Dividing the weight of the piece by the weight per 
cubic inch gives the volume, which divided by the length gives the area ot the 
cross-section (.002032 sq. in.). Dividing the weight of the piece by its length 
gives the weight per linear inch (.000574 lb.). Multiplying the area of cross-sec¬ 
tion by 40 000 shows that a pull of over 80 lbs. will be required to strain the tape 
40 000 lbs. per square inch, which is probably within the elastic limit. 


a 

E 


0.01959 

30 165 000 


0.01949 

30 330 000 


0.01932 

30 597 000 

/ 

0.02878 

30 810 000 

- 

0.02928 

30 283 500 


0.03857 

30 652 500 

















86 


for tlie difference in tension l u — h = -f- $2 — t>i 

= (Vn — ®i) — ( Si — « a ) = (^a — Vi) ~ C 8 . 

rp _ p \p 

The Modulus of Elasticity E = 1 , a ^ in which 5 is the 


cross-section of the tape in square inches (0.00203). 

As a sample computation, C s = ^ (.000574 X 360) 2 
= 0.00128. Then (® a - Vi ) - C s - 0.02087 - 0.00128 = 0.01959 = a 


(in feet). E = 


(16 - 12)3600 


0.00203 X (0.01-959 X 12) 


= 30 165 000, 


— .35 — 

AZIMUTH, USING SOLAR ATTACHMENT.* 

Equipment : Transit, with solar attachment ; set of declination 
tables. The vertical circle of the transit should b a fixed and have a 
vernier reading to half-minutes. 

Location : This problem must be worked with the instrument at 
some point from which a clear view of thesuu, unobstructed by 
trees, buildings, etc., may be obtained throughout tlie whole period 
of the work. The problem consists in finding the true azimuth of a 
line from the instrument (which should be set over a fixed hub) to 
some prominent mark (e.g., a distant steeple) which is suitable as 
an azimuth-mark. 

Time : From two to four hours before or after noon. 

Method : Compute the declination-settings* for each half-hour for 
the total period of the observations. Record these at once in the 
note-book, leaving an interval of about five lines for observations 
made during each half-hour. The ten desired observations w 7 ill not 
generally require much more than one hour and with practice ten 
observations may be made in a few minutes. 

Adjust the eyepieces of both telescopes for parallax, focus the 
object-glass of the solar for observing the sun and focus the object- 
glass of the transit-telescope for observing the azimuth-mark. 

Level up with extreme care. The plate-bubble parallel to the tele¬ 
scope is almost invariably too coarse for the accuracy required in 


* See the discussion on Azimuth in Appendix B, p. 90. 





87 


this work. Clamp the telescope nearly horizontal. By repeated re¬ 
versions of the whole instrument about its vertical axis, with corre¬ 
sponding adjustments of the levelling-screws and the tangent-screw 
to the vertical arc, the instrument may be so levelled that the telescope 
level-bubble will remain in the centre of the tube for any position of 
the instrument. Under these conditions the vertical axis is truly 
vertical and the reading of the vertical arc-will be its index error- 
assuming that the line of collimation is parallel to the axis of the 
bubble-tube. The exact error of the adjustment of the plate-bubbles 
is then apparent and should be noted, so that any accidental change 
of level that may occur during the time of taking observations may 
be at once detected. 

Set vernier A of the horizontal plates at 0° and point at the azi¬ 
muth-mark. Loosen the upper plate and swing the telescope ap¬ 
proximately into the meridian, the telescope^ pointing south. If the 
declination is south, point the transit-telescope upward with a 
vertical angle equal to the value of the declination as modified by 
refraction. If the declination is north, point the telescope down¬ 
ward with a vertical angle equal to the modified declination.* Then 
place the solar telescope in the same vertical plane as the transit-tel¬ 
escope and make it horizontal by means of its level-bubble, clamping 
it securely. Then make the transit-telescope parallel to the plane 
of the equator by pointing it upward with a vertical angle equal 
to the co-latitude. The polar axis of the solar is thus made 
approximately parallel to the earth’s axis ; it will be exactly parallel 
when the transit-telescope is exactly in the meridian. Since the me¬ 
ridian is obtainable to 'within a small range by means'of the mag¬ 
netic needle or by an approximate solar observation, the most con¬ 
venient method is to bring the transit-telescope into the meridian 
as nearly a3 possible and clamp the upper plate. Point the solar 
telescope at the sun by the shadow of the sights on top and clamp 
the motion about the solar axis. Then, with one hand on the slow- 
motion screw of the polar axis and the other hand on the tangenl- 
screw of the upper transit-plate, point the solar telescope exactly at 
the centre of the sun. The reading of the horizontal plate gives one 
value of the angle between the azimuth-mark and the meridian. 
To obtain another value which shall be independent of the previous 

♦ To find the modified declination, see the discussion on Azimuth in Appen¬ 
dix B, p. 90. 





88 


determination, loosen the upper transit-plate and the clamp-screws 
of the solar and repeat the operation, making due allowance (if 
necessary) for any change of declination that may have taken place 
in the interval. With practice several observations may be taken 
in a very few minutes, during which time no appreciable change of 
declination will take place even when the motion of the sun in 
declination is most rapid. Watch the levels carefully for any indi- 


Form of Notes. 


Time. 

Decl. 

Refr. Corr. 

App. Decl. 

Angle between 
Azimuth-mark 
and Meridian. 

Needle. 

3:54 

+ 0° 50'.4 

4-1'.5 

4-0° 51'.9 

41° 21' 

N. 7° 30' W. 

57 



51 .9 

19 


4.02 

-f 0 50 .3 

4-1 .6 

4-0 51 .9 

19 


07 



51 .9 

20 


14 



51 .9 

19 


17 


• 

51 .9 

19 


19 

+ 0 50 .0 

4-1 .9 

4-0 51 .9 

19 


22 



51 .9 

18 


26 



52 .0 

18 


4:30 

4-0 49 .8 

4-2 .2 

4-0 52 .0 

41 19 






41° 19'. 1 



- 36 - 

azimuth, FROM ALTITUDE AND DECLINATION OF 
THE SUN, USING TRANSIT.* 

Equipment: Transit, provided with vertical circle, which should, 
if possible, be graduated to 80" of arc. A colored glass shade is also 
necessary to protect the eye : this glass may be placed either over the 
eyepiece or over the object-glass, but if placed over the object-glass 
it must be optically correct, i.e., the surfaces perfectly plane and 
parallel to each other so that there is no distortion or refraction of 
the light-rays passing through. When the transit has only a semi¬ 
circular arc, or a movable arc which is set by clamping, Method 
A must be used. When the instrument is equipped with a full 
circular arc, Method B will be found more accurate. 

Location: [See directions for location in Problem 35.] 

Time : From two to four hours before or after noon. 


* See discussion on Azimuth on pp. 90 et seq.' 

















89 


cation of jarring of the instrument. The lower plate should be 
kept clamped throughout and the vernier should always read 0° 
wheu pointiug at the azimuth-mark. Observe the needle reading 
when the instrument is in the meridian. It will show the declination 
of the needle foi that time and place. 

Take ten observations and compute the probable error of the mean 
and of a single observation. 


d 

d* 


1.9 

3.61 

Philadelphia, Sept. 20, 1898. . 

0.1 

.01 

5 (at noon at Greenwich). 

0.1 

.01 

8 hours change at — O'.97. 

0 9 

.81 

8 (3 p.m., Phila.). 

0.1 

.01 

8 (4 p.m.). 

0.1 

.01 

8 (5 p.M.). 

0.1 

.01 


1.1 

1.1 

1.21 

1.21 

Ei = .6745 V ^ = ± O'.59 

0.1 

.01 

0.59 


6.90 

E 0 = —= = ± O'.187 

yio 



= ± 11 ".20 


</> = 39° 57 
+ 0° 59'.0 
7 .76 


4-0° 51'.24 
-fO 50.27 
+ 0 49.30 


— 36 CONTINUED — 

Method A. 

Method: The problem consists in obtaining the altitude of the 
centre of the sun above the horizon and also the horizontal angle 
between the sun’s centre and some definite mark at the same instant 
of time—which must be observed. Then, knowing the altitude and 
declination of the sun at that instant and also the latitude of the 
place of observation, the angle between the sun’s centre and the 
meridian is computable, and then by mere addition or subtraction 
the azimuth of the mark from the place of observation is obtained. 

Set up the transit over the designated hub and level up with 
especial care (using precautions suggested in Problem 35), since the 
accurate determination of vertical angles is necessary. Point at the 
azimuth-mark with the horizontal plates reading 0°. With the upper 
plate loose, point at the sun, observing the time, altitude, and the 
horizontal angle from the azimuth-mark. Take six observations, 















90 


which will, of course, be different, as the altitude and azimuth are 
constantly changing. Finally, point at the azimuth-mark to test 
whether the lower plate has slipped. The reading on the azimuth- 
mark should be 0°. 

Pointing at the sun. The sun’s angular diameter is about 0° 32'. 
With the comparatively high power telescopes now generally used 
on transits, this fills a large part of the field of 
view and it is impossible to accurately bisect such 
a large angular width, especially as the apparent 
motion of the sun across the field of view is very 
rapid. It therefore becomes advisable to sight 
the cross-wires on the edges of the sun, as shown 
in the figure, and make due allowance for the 
semi-diameter of the sun. The effect of this is to obtain an altitude 
which differs from the true altitude (h) by the angular value of the 
semi-diameter. The observed azimuth differs from the true azimuth 

by the sem ^~^^ a ^ ete J < "When the sun is at the horizon, cos h = 1, 
J cos h 

and the allowance equals the semi-diameter both for altitude and 
azimuth. For high altitudes the allowance for azimuth is much 
larger than the semi-diameter, since the divisor (cos h) is small. If 


Form of Notes.* 


Time. 

Apparent 

Altitude. 

a 

h 

5 

Z 

4:50 p.m. 

22° 48'.5 

237° 41' 

22° 30'.3 

14° 45'.6 

89° 16'. 6 

4:53 

22 12 .5 

238 11 

21 54 .3 

45 .6 

88 46 .6 

4:55 

21 44 .5 

238 34 

21 26 .2 

45 .6 

88 23 .3 

4:58 

21 19 .0 

238 55 

21 0 .7 

45 .7 

88 02 .4 

5:00 

20 49 .5 

239 19 .5 

20 31 .1 

45 .7 

87 38 .0 

5:03 

20 28 .0 

239 38 

20 9 .5 

14° 45 .7 

87 19 .9 


* These figures were obtained by one of the author’s students and represent 
his first attempt at this kind of work. 


In the Form of Notes the column-headings signify as follows: 

a = horizontal angle from the azimuth-mark, the angle being measured to the 
right. 

h — apparent altitude — refraction ± semi-diameter of sun, in vrhich 
semi-diam. is -f- when sun is above the horizontal cross-wire, 

“ « “ - “ “ “ below “ “ “ “ • 

S = declination. 

Z — computed angle (as illustrated below). 
















91 


several observations are taken within a short interval, the change in 
this allowance for azimuth may be too small for notice and one value 
may be sufficiently accurate for all the observations. It will probably 
be found easier to obtain simultaneous contact of both wires by using 
the lower left-hand (or upper right-hand) 
corner of the field of view for morning work 
and the upper left-hand corner (or lower 
right-hand corner)for afternoon work. There 
is a slight variation in the semi-diameter, as 
is shown by the accompanying tabular form 
giving average values, which may be used 
by interpolation if closer values are desired. 

Latitude: The latitude of the place of observation to the nearest 
half-minute of arc is necessary. This is generally known or may be 
obtained as shown in Problems 87 and 38. 

Reducing the observations. Compute the declinations for the given 
times of observation. If several observations are taken, it is generally 
sufficiently accurate to compute the declinations for the times of the 
first and last observations and interpolate for the others. The 
observations may be most readily reduced by using a regular form, 
as given below. 


Time 

of 

Year. 

Semi-diam. of 
the Sun in 
Min. of Arc. 

Jan. 1 
Apr. 1 
July. 1 
Oct. 1 

16'. 31 (max.) 
16 .03 

15 .77 (min.) 

16 .03 


semi-diam. 

True Azim. 
of Mark. 

• 

cos app. alt 

17'. 2 

17 .2 

17 .1 

17 .1 

17 .0 

17 .0 

213° 19'. 6 
19 .6 
19 .8 
19 .7 
19 .5 
213 19 .1 

April. 29, 1897 
<f> = 39° 58' 


semi-diam. sun = 15'.9 

- 


213° 19'.55 = mean value. 


. - , _semi-aiam. _ . . , 

True azimuth of mark = 540° ± -r— ± Z - a, in which Z is -f for a.m. 

COS Q.lt. 

and — for p.m., and the 

semi-diam . . g w h en the sun is on the left of the middle wire (as above), 
cos alt. 

(I 44 _ 44 44 44 44 44 44 yigjlf “ “ “ “ 

If “true azimuth of mark” is computed to be more than 360°, subtract 360° 
from it. 


















92 


As a numerical example of the reduction : App. deol. Greenwich mean noon 
April 29, 1897, -f 14° 38'.0; hourly change-f O'.77; difference of time between 
Greenwich and Philadelphia 5.0 hours; 5 p.m. at Philadelphia is 10 p.m. at Green¬ 
wich; therefore 5 for 5 p.m. at Philadelphia = -f- 14° 38'.0 -f- (10 X O'.77) = -f- 14° 
45'.7. From Eq. 1 (see article on Azimuth, p. 88.) 


sin = 


7 


sin (s — co. li) sin (s — co. 
sin co. h sin co. 4> 


±). 


co. h = 67° 29'.7 
co. <f> = 50 02 .0 
co. S = 75 14 .4 
192° 46'. 1 
8 = 96 23 .0 


s — co. h = 28° 53'.3, sin = 9.684041 
s — co. </> — 46 21 .0, sin = 9.859480 

9.543521 

sin co. h = 9.965599 
sin co. <}> = 9.884466' 

9.850065 9.850065 

2 | 9.693456 

— 9.846728 = sin 44° 38'.3 


= 44° 38'.3; .\ Z = 89° 16'.6. , 

semi-diam. sun _ 15.9 _ 2 

cos app. alt. ~ cos 22° 48' — 

540° 4- 17/2 = 540° 17'.2 

- Z - a = - 89° 16'.6 - 237° 41' = - 326 57'.6 
True azimuth of mark = 213° 19'.6 


Method B. 

Method: The method differs from the preceding in that the obser¬ 
vations are made in pairs, the telescope being plunged and the upper 


Fokm of Notes for Method B. 


Time. 

Apparent 

Altitude. 

a , 

Refraction. 

h 

£ 

( 3:27 

19° 39' 


99° 

52' 










J 3:29 

19 52 


99 

49 










(3:28 

19° 45' 

o 

CO 

99° 

50' 

30" 

2' 

40" 

19° 

42' 

50" 

-9° 

o 

CO 

00" 

(3:32 

18° 46' 


>-* 

O 

O 

O 

55' 

00 

O 









j 3:34 

19 03 


100 

49 










(3:33 

18° 54' 

30" 

100° 

52' 

15" 

2 

48 

17 

51 

42 

-9 

30 

00 

(3:36 

18° 04' 

30" 

101° 

46' 










J 3:38 

18 23 

30 

101 

35 










(3:37 

- 

00 < 
O 

■ 


101° 

40' 

30" 

2 

54 

18 

11 

06 

-9 

30 

00 

(3:40 

17° 26' 

30" 

102° 

29' 

30" 









! 3:42 

17 43 


102 

21 










(3:41 

17° 34' 

45" 

o 

Cl 

o 

25' 

15" 

3 

0 

17 

31 

45 

-9 

30 

15 





























93 



plate swung 180° in azimuth between each observation. This has 
the effect of eliminating all index error of the vertical circle, error 
of collimation, and error due to difference of 
height of the telescope-standards. The instru¬ 
ment must be equipped with a complete ver¬ 
tical circle. Then, if the observations of a pair 
are taken in diagonal corners of the field of view 
(as shown in the figure), even the correction for 
semi-diameter of the sun is eliminated. Ob¬ 
serve as before the time, altitude, and azimuth. 

Correct the altitude for refraction. Add (or 

subtract) 180° to the reading of vernier A for the readings when the 

telescope is inverted. 

Compute a mean Z for each pair of observations as before except 
that a = the mean horizontal angle, 

h = the mean altitude — refraction, 

8 = the declination for the mean time, and 
True azimuth of mark = 540° ± Z — a. 

Tak efour pairs of observations. 


V_ 


z 

True Azim. of 
Mark. 

* 



October 17, 1898 



<J> = 39° 57' 

121° 55' 30" 

318° 14' 00" 


120 54 30 

318 13 15 


120 05 30 

318 14 00 


119 22 00 

318 12 45 



318° 13' 30" 














94 


Method C. 

The two preceding methods have the disadvantage that the results 
of the field-work require considerable calculation before they have 
any value or before any idea of their accuracy may be obtained. 
The following method gives the meridian directly, but has the dis¬ 
advantage that the observation must be taken at some previously 
calculated instant of time, at which time the sun may be obscured 
by clouds or the observation is unobtainable for any one of many 
reasons. But when the weather may be depended on and sufficient 
precautions are taken so that nothing interferes with taking an ob 
servation at the required instant, the method is perhaps the best of 
any of the solar methods. 

Method: Calculate from formula (2) (see Appendix, p. 92) the 
altitude (h) of the centre of the sun for a given time and date. Then 
calculate from formula (1) the angle Z corresponding to that altitude. 
By this method the horizontal angle from the sun to the meridian at 
any chosen instant becomes known. Setting off this horizontal angle 
and altitude a few minutes before the given time, move the whole 
instrument in azimuth until the telescope is pointing directly at the 
sun. A 0° reading of the horizontal circle should then give the true 
meridian. Then sight on the azimuth-mark and read vernier A, 
reading the row of degree numbers that runs to 360°. Since the 
azimuth readings and the altitudes are different for each observation, 
the readings on the azimuth-mark are independent observations on 
the value of the required angle. 

Form of Notes for Method C. 


Apparent 

Time. 

h 

Refraction. 

h-f semi-diam. 
refraction. 

Z 

Semi-diam. 
cos alt. 

9:00 

26° 20' 54" 

1' 55" 

26° 38' 54" 

128° 40' 00" 

18' 00" 

05 



f27 22 36 "1 



10 



L 28 06 18 J 



9:15 

28 32 10 

1 45 

28 50 00 

132 02 30 

18 21 

20 



f - 29 31 22 q 



25 



|_30 12 44 J 



9:30 

30 36 24 

1 36 

30 54 05 

135 34 50 

18 41 














95 


As explained in Method A, the cross-wires of the telescope should 
preferably be pointed at the edges of the sun, which requires that the 
altitude should be corrected by the serai-diameter of the sun (as well 
as by the refraction) and that the azimuth should be corrected by the 

—- f - m - * ’ a m ',-, Since more accurate work may be done when the 
cosine app. alt- 

sun is apparently moving toward the wires, the telescope should be 
so pointed that the sun is in the lower left-hand corner of the field 
of view in the morning and in the upper left-hand corner in the after¬ 
noon. The tiltitude and azimuth set off should be correspondingly 
altered. 

The hour-angle (t) is the angle from the true meridian and differs 
from standard time and even-from rttean local time. The exact time 
is unimportant in this problem, it being only necessary to know ap¬ 
parent time with such accuracy that the instrument may be properly 
set a few seconds before the time for taking the observation. 

Make calculations for three observations 15 minutes apart and in¬ 
terpolate so as to take four intermediate observations. 

Since errors of altitude produce magnified errors in the resulting 
azimuth, the quantities are computed to seconds of arc. In setting 
off the angles the closest possible setting to the real value should be 
made. The angles [180° — Z — (semi-diam. -r- cos alt.)] should be 
set off on the left side of the 0° in the forenoon and on the right side 
in the afternoon, noting carefully, however, the algebraic sign of 
(semi-diam. cos alt.). # 


180° - Z 

semi-diam. 
cos alt. 

Azimuth of 
Mark. 


51° 02' 00" 

318° 11' 

October 14, 1898 

r49 54 23 1 

318 13 

</> = 39° 57' 

l_48 46 46 J 

318 12 

Semi-diam. of sun 16' 05" 

47 39 09 

318 14 



r46 28 16 1 

318 12 



L45 17 23 J 

318 11 

0 


44 06 29 

318 14 



Mean = 318° 12' 26" 













96 


— 37 — 

LATITUDE, FROM C1RCUMMERIDIAN ALTITUDES OF 
THE SUN, USING TRANSIT.* 

Equipment: Transit, provided with a vertical arc and a colored 
glass shade to protect the eye. [Note the required equipment for 
Prob. 36.] 

Location: [See directions for Location in Prob. 35.] 

Time: The transit should be set up.so that observations can be 
commenced at about 15 minutes before “apparent” noon, which is 
not “standard time ” noon nor even “ mean local time ” noon, but is 
the time that the sun actually crosses the meridian, which differs 
from “mean local noon” by the “ equation of time.” If a solar 
ephemeris is available and if true mean local time is accurately 
known, the precise instant of apparent noon is readily obtainable and 
the reductions are simplified—as shown below. A rough approxi¬ 
mation of the time of apparent noon may be found by interpolating 
in the tabular form. 


Date. 

Mean Local Time of 
Apparent Noon. 

Jan. 1 




12 h. 

3 m. 

50 s. 


Feb. 10 

(approx.) 

12 

14 

26 

(max.) 

Mar. 15 




12 

09 

0 


Apr. 15 

( 


) 

12 

0 

0 


May 14 

( 

41 

) 

11 

56 

10 

(min.) 

June 14 

( 

44 

) 

12 

0 

0 


July 26 

( 

(4 

) 

12 

06 

17 

(max.) 

Aug. 31 

( 

4 4 

) 

12 

0 

0 


Oct. 1 




11 

49 

35 


Nov. 3 

( 

44 

) 

11 

43 

40 

(min.) 

Dec. 1 




11 

49 

15 


Dec. 24 

( 

44 

) 

12 

0 

0 



Method: Level up the instrument with extreme care, using pre¬ 
cautions suggested in Prob. 35. At about 15 minutes before appar¬ 
ent noon sight the horizontal cross-wire on the upper edge of the 
sun, noting the time of exact contact and then reading the vertical 


* The method here developed is based on that used in refined astronomical 
calculations, but is very greatly simplified by approximations, which, however, 
do not cause errors as great as T \, of a minute of arc, which is far within the 
limit of accuracy of an ordinary engineer’s transit. 







97 


angle. Repeat these observations as rapidly as is consistent with ac¬ 
curate work and continue the observations until about 15 minutes after 
noon. 

Reduction of observations. An observation at apparent noon will 
have a greater altitude than an observation before or after noon— 
disregarding the effect of change of declination, which will hardly 
be appreciable with an ordinary engineer’s transit. Observations 
taken before and after noon can therefore be compared with one at 
noon by adding to them a computed correction, which consists of a 
series of terms of which only the first need be taken. This correc- 

2sin* U cos (p cos 5 . . , ., << •- ^ 

tion x =-- ———- T7, in which t = the hour-angle, ep = 

sin 1 sin ((p — o) 

latitude, and 8 = declination. The value of t is found as shown be¬ 
low ; cp may be found with sufficient accuracy for the purpose from 
the equation ep = 90° - (A - semi-diam. — 8), in which A is the 


cos (p cos 8 

maximum altitude obtained during the observations. ^ —~8 ) 

2 sin* 

is sensibly a constant for any one set of observations, while g . Q y, 


depends only on the hour-angle from the meridian, and the values 
for it are tabulated for each 10 seconfis of time up to 20 minutes from 
the meridian in Table XI, A value for (p is obtainable from each 
observation by the formula 


fp = 90° — (A — refrac. — semi-diam. — 8 -f x). 


Hour-angle ( t ). This is the angular value of the difference of 
time between the time of the observation and the time of apparent 
noon. When the observer is provided with accurate mean 
local time, he may obtain the time of apparent noon as 
illustrated numerically below; otherwise the time of apparent 
noon may be obtained with sufficient accuracy for this purpose 
by plotting the altitudes on profile-paper, plotting differences of 
time at the scale of 8 minutes of time per inch horizontally, and 
differences of altitude at the scale of 2 minutes of arc per inch verti¬ 
cally. The points should plot into a parabolic curve with vertical 
axis, which axis will denote the time of apparent noon. If an obser¬ 
vation was taken exactly at apparent noon or at culmination, it would 
be plotted at the vertex of the parabola and this observation would 
have no correction ( x ). The time need only be taken to the nearest 






98 


even 10 seconds. As an illustration, the observations recorded in the 
Form of Notes have been plotted as shown. The vertical circle 



used was graduated to single minutes and estimated to half-minutes. 
The coarseness of these^readings will partially account for the varia 


Form of Notes. 


Time. 

Altitude. 

Altitude 
Corrected for 
Refr. and 
Semi-diam. 

Hour-angle 

it). 

Correction 

(x). 

Altitude 
Reduced to 
the Meridian 

llh. 33 m. 20 s. 

3:° 21' 30" 

37° 04' 

05" 

9 m. 

10 s. 

2' 

33 ' 

37° 

06' 38" 

35 

0 

22 0 

04 

35 

7 

30 

1 

42 


06 17 

36 

10 

22 30 

05 

05 

6 

20 

1 

13 


06 18 

38 

0 

23 0 

05 

35 

4 

30 


37 


06 12 

39 

30 

23 0 

05 

35 

3 

0 


17 


05 52 

42 

30 

23 30 

06 

05 

0 

0 


0 


06 05 

44 

0 

23 30 

06 

05 

1 

30 


4 


06 09 

46 

30 

23 30 

06 

05 

4 

0 


29 


06 34 

48 

0 

23 0 

05 

35 

5 

30 


55 


06 30 

52 

0 

21 30 

04 

05 

9 

30 

2 

44 


06 49 

53 

30 

20 0 

02 

35 

11 

0 

3 

41 


06 16 

11 55 

30 

18 30 

37 01 

05 

13 

0 

5 

18 

37 

06 23 

Means .. 


37° 22' 07" 





r 

38" 

37° 

06' 20" 

Mean x .. 


+ 1 38 






6 = 

12 

58 0 



CO 

o 

8 

U* 





co-6 = 

50° 

04' 20" 

Refr. 4- 











semi-diam. 

17 25 






<*> = 

39 

55 40 

Mean alt. 


37° 06' 20" 











































99 


tions between the plotted points and the curve. The reliability of 
this method (in the absence of an Ephemeris) is shown by the fact 
that although the ordinate at 11 h. 39 m. 80 s. is evidently very far 
from the vertex of the curve, yet if the “cc” corrections were com¬ 
puted on the basis of apparent noon occurring at that time instead 
of at 11 h. 42 m. 30 s. the final result would be altered only 3" of 
arc, which is far within the lowest unit of these observations. 

It should be noticed that the final result (the mean value of <p) 
may be obtained from columns 1, 2, 4, and 5, without reducing each 
separate observation, by employing the mean altitude and mean x. 
The work is thereby much shortened, but it has the disadvantage of 
giving but little idea of the probable error of the result. A mere 
inspection of the independent results given in column 6 shows the 
accuracy of the work except as it may be affected by constant errors , 
such as errors of adjustment. 


October 22, 1898. Upper limb of sun. 


Max. alt. = 37° 06' 05" 

— (— 6) = + 12 58 0 

50° 04' 05" 


Approx. (f> = 39 56 .cos = 9.8847 

5 for Gr. noon = 12° 53'.8 

5 X 0.34 = 4 .2 

5 — 12° 58' .0.cos = 9.9888 

9.8735 

_ (_ S) = 52° 54'.sin = 9.9018 

Coeff. for (x) = 0.937 .9.9717 


Refr. = 1' 16" 

Semi-diam. sun= 16 09 

17' 25" 

Eq. of time, — 46 m. 03 s. 

Mean local time of app. noon, 11 h. 43 m. 57 s. 
Watch slow (mean local time), 1 m. 20 s. 
Watch time of app. noon, 11 h. 42 m. 27 s. 


LOfc. 















100 


— 38 - 

* SEXTANT PRACTICE—HORIZONTAL ANGLES. 


Equipment: Sextant. 

Select four well-defined points at least ^ mile away (e.g., chim¬ 
neys, steeples, etc.). Obtain four values for each of the six angles 
formed at the instrument by the points taken in pairs, as shown 
below. Loosen the clamp and move the arm slightly between each 
reading. Unless the points sighted at are very far away, it will be 
necessary to use considerable care in holding the sextant over a fixed 
point or an appreciable variation in the angle may be the result. 
This problem is useful as preliminary practice for the work of locat¬ 
ing soundings by means of sextant-angles taken from a boat. 


Form of Notes. 


Angle. Readings. Mean, 


(similarly) 


BOO 


AOB 


COD 


12° 1G' 20" 


14 22 40 


etc. 


10 

30 

20 12° 16' 20" 



0 

Church steeple, cor. E. and F. sts. 


Id 


AOC 


A, Church steeple, cor. E. and F 

B, N.E. cor. G. and H. sts. 

C, Tower, City Hall. 

D, S.W. cor. M. and N. sts. 
Sextant at N.E. cor Chem. Lab. 


BOD 


AOD 


[The student should place the sketch on a right-hand page of his note-book 
the computations on a left-hand page.] 

















101 

— 39 — 

SEXTANT PRACTICE—TESTING ADJUSTMENTS * 

Equipment: Sextant; also two blocks of wood about in height. 

1. The plane of the index-glass should be perpendicular 
to the plane of the arc. The effect of an error in this adjust¬ 
ment is not readily capable of measurement. Set the index-arm 
at about 60° or 70°; hold the sextant nearly horizontal with the arc 
away from the eye ; observe in the index-glass the reflection of the 
arc and note whether the arc as seen directly forms a continuation 
on both sides of the reflection seen in the index-glass. 

2. The plane of the horizon-glass should be perpendicular 
to the plane of the arc. An accurate test of this adjustment 
requires that the telescope should be pointed at a star, and it can 
be satisfactorily tested only when a distant and very sharply de¬ 
fined object may be sighted at. A finial, seen against a bright sky 
background, will often prove to be the best terrestrial mark obtain¬ 
able. Use the highest power telescope that belongs to the sextant 
for this test. Set the index-arm at 0° and sight at the point ; the 
direct and reflected images will nearly coincide. Move the index- 
arm slightly in either direction. If the sextant is held horizontally 
and the horizon-glass is not in adjustment, the reflected image of a 
point will be seen to pass the direct image either above or below it. 
A rough measure of the value of such a lack of adjustment may be 
obtained by reading the vernier when the direct and reflected images 
are directly above or below each other (the sextant being horizontal), 
and again reading it when the reflected image is as far (by estima¬ 
tion) to the right or left of the direct image as it is above or below 
it. The first of these readings should be 0° if there is no “ index 
error.” If more convenient, the sextant maybe held vertically, in 
which case the reflected image will pass to the right or to the left of 
the direct image. The difference of the above readings will be four 
times the lack of perpendicularity of the glass. The effect on meas¬ 
ured angles of a small error of this sort cannot be detected except 
with the closest work. Obtain three differences as above described 
on each side of the point where the images seem most nearly in coin¬ 
cidence. 


* See first paragraph of Problem 12, page 16. 





102 


3. The axis of the telescope should be parallel to the plane 
of the arc. Set the sextant horizontally on a firm support which 
is 15 to 20 feet from a wall. Place two blocks on the arc so that a 
sighting may be taken over the blocks to the wall at the place at 
which the telescope points. The height of the blocks should be the 
same and equal to the height of the centre of the telescope above 
the arc. The height of the telescope may be adjusted to the exact 
height of the blocks if the telescope is adjustable. Sight over the 
tops of the blocks and have an assistant make a mark on the wall. 
Then, without disturbing the instrument, note whether the telescope, 
previously focussed and directed to that place on the wall, is sighting 
above or below T the mark. Make three trials. 

4. To determine the index error. Although all sextants are 
provided with adjusting-screws for correcting this error, it is consid¬ 
ered best to be content with making it small and carefully computing 
its amount, which should then be applied to all readings. Sight at 

Form of Notes. 


Adjustment. 

. 

1st Trial. 

2d Trial. 

3d Trial. 

Mean. 

1 






3 

Reft. im. above. 
“ “ oppos. 

“ “ below. 

- V 30" 
4 1 o 

4- 3 10 

-O' 50" 
4 1 50 
4-3 30 

- 1* 0" 
4- 1 20 
+ 3 0 



Diff. above .... 

‘ ‘ below . .. 

2' 30" 

2 10 

24' 0" 
14 0 

2' 20" 

1 40 

2' 30" 

1 50 

3 


+ 2W' 

4-2%" 

+ 2%" 

+ 2^" 



On Arc. 

Off Arc. 

H Diff. 


4 


33' 40" 

50 

60 

40 

60 

?5 

60 

70 

80 

75 

30' 40" 

15 

20 

40 

30 

50 

50 

15 

10 

35 





610" 

34' 01" 

305" 

30' 30".5 

- 1' 45".2 






































103 


a well-defined point with the index arm at or near 0°. Bring the 
two images into exact coincidence. The arc is graduated for a few 
degrees back of 0°. If the zero of the vernier is back of 0°, or “off 
the arc,” all readings of angles will be too small and the correction is 
positive. If the zero of the vernier is “ on the arc,” the correction is 
negative. As shown in No. 2, a terrestrial object is not sufficiently 
well defined for accurate work. At night a star gives the best 
mark. In the daytime the sun may be sighted at, but, since it is 
very difficult to bring the two images of the sun into accurate coin¬ 
cidence, it is best to make the images tangent. Read the vernier. 
Then move the index arm until the opposite limbs are tangent aud 
again read the vernier. One reading will be off the arc and the 
other on the arc. One-half of the difference equals the index error, 
and its sign will be according as the reading on the arc or off the 
arc is greater. Obtain 10 readings on the sun off the arc and 10 
readings on the arc. 


No perceptible error. 


General mean 2' 10"; error of glass 32".5. 

Wall 35' from sextant. Telescope points above the arc. 


Sighting on sun. 


Mean reading is on the arc. Therefore the correction is negative. 














104 


— 40 — 

LATITUDE FROM CIRCUMMERIDIAN ALTITUDES OF 
THE SUN, USING SEXTANT. 

Equipment: Sextant, artificial horizon, chronometer or watch. 
[For Location, Time, and Reduction of Observations see Prob. 87.] 
Method: The method is nearly identical with that of Prob. 37 


Form of Notes. 


Time. 

Double 

Altitude. 

Hour-angle 

(Ph 

Correc. to 
Alt. (a;). 



11:39 

OO 

o 

O 

qj 

10" 

15 m. 30 s. 

504" 



42 

10 

40 

12 

30 

328 



44:30 

13 

10 

10 


210 



47 

16 

00 

7 

30 

118 



48:40 

17 

50 

5 

50 

71 



50:30 

19 

40 

4 


34 



53:30 

20 

40 

1 


2 



55:30 

20 

40 

1 


• 2 



57:20 

19 

30 

2 

50 

17 



59 

19 

50 

4 

30 

43 



12:01:10 

16 

50 

7 

20 

113 



03:20 

13 

20 

8 

50 

164 



05:30 

12 

30 

11 


254 



07 

89 08 

00 

12 

30 

328 




89° 15' 

16" 



156" 



Index error 

— 1 

45 



2' 36" 



2[ 

89° 13' 

31" 







44 36 

45 






Corr. (x) 

4- 2 

36 






Refr. 


59 







44° 38' 

22" 






Semi-diam. 

- 16 

03 







44° 22' 

19" 






- (- *) = 

+5 41 

50 






CO <f> 

50° 04' 

09" 






<*> 

39 55 

51 


' 


1 





















105 


except that the double altitude is observed, since an artificial horizon 
is used. Also, as shown in Prob. 37, the observations need not be 
reduced separately, but the means may be used after the corrections 
(a;) have been computed. 

To obtain the altitude of the upper limb, bring the two images in 
contact so that the direct reflection (from the artificial horizon) is 
above the index-image. Of course the inversion of an inverting tele¬ 
scope, if used, must be allowed for. Obtain the altitude of the lower 
limb by bringing the direct reflection below the index-image. 


Oct. 7, 1898. Upper limb of sun. 

Approx, long. = 75° 16' 30". 16' 30" = 1 m. 06 s. 

Eq. of time = 12 m. 12 06 s. 

5 X 0.71 =_ 3.55 s . 

Corr. eq. of time = 12 m. 15.61 s. * 
Apparent noon occurred at 11 h. 47 m. 44 s. mean local time 
or at 11 h. 54 m. 30 s. watch time, 
the watch being 5 m. 40 s. fast of 75* time 

and 6 m. 46 s. “ “ 75° 16' 30" time. 

Max. double alt. = 89° 20' 40" 

Corr. for index error = 89 18 10 
Altitude = 44 39 05 

Corr. for semi-diam. = 44 23 

(Approx.) S = 5 42 cos S = 9.9979 

co <f> 50° 05' ' cos <#> = 9.8848 

(Approx.) <f> 39 55 9 8827 

(f> — (— 8) = 45 37 sin (<f> — S ) 9.8540 

Factor for ( x ) = 1.07 . . . 0.0287 









106 


- 41 - 

time AND LONGITUDE. 

[The methods here given are introduced. solely as preliminary 
practice to the more accurate astronomical methods. The assumed 
lack of a well-rated chronometer at once precludes the possibility of 
great accuracy.] 

Equipment: Transit with vertical circle (or sextain and artificial 
horizon), and a watch having a seconds-hand. 

Location: Preferably at a station at which Latitude and Azimuth 
have already been obtained. 

Time : At noon for Method A ; at from three to five hours from 
the meridian for Methods B and C. 

Method A. Using Transit, at Noon. 

This method assumes the previous determination of azimuth, so 
that the true meridian is known. Set the telescope in the true 
meridian and note the time of passage of the east and west limbs of 
the sun past the vertical cross-wire. The mean of these times gives 
the time of the passage of the sun’s centre across the meridian, but 
this gives only a single unchecked determination. The Ephemeris 
gives the time required for the sun to pass the meridian for each 
day of the year, the time varying from 64 to 71 seconds. By add¬ 
ing (or subtracting) one-half of this period to the time of passage of 
each limb two independent observations are obtained. The reduc¬ 
tion of the observations will be clear from the example given 
below. 

An error of 1 second of time will cause an error of 15" of longi¬ 
tude ; an error of 1' in the setting of the telescope in the meridian 
will cause an error in longitude somewhat less than 1', the amount 
depending on the altitude and declination; an error in the heights of 
the telescope-standards will also introduce an error which is greater 
as the altitude is greater. The last source of error may be elimi¬ 
nated by reversing the telescope between the two observations. A 
determination of longitude much within 30" of arc is therefore 
doubtful by this method. 

The accuracy of all these methods may^be increased by having 
the observer call out “ ready ” to an assistant a few seconds before 


107 


contact takes place; the assistant counts seconds and, at a call of 
“ tip ” from the observer, notes the time, estimating tenths of a 
second. 


Method B. Using Sextant, at Any Time. 

j Forenoon observations: Obtain first the approximate double alti¬ 
tude of the upper limb of the sun and then set the index at an even 
20' mark which is about 20' greater than the observed approximate 
double altitude. Watch the two images approach each other and 
note the exact time of contact. Then increase the reading by an even 
20' and observe the time of contact, as before. After obtaining five 
such readings reduce the double altitude 40' and observe the time of 
contact of the images of the lower limb. The images will be seen to 
overlap and to be separating. Increase the readings by the uniform 
difference of 20' and take Jive readings, as before. Afternoon obser - 
vations: Observe the lower limb first, in which case the images are 
approaching each other ; then observe the upper limb, in which 
case they overlap and are separating. The successive readings 
should then be decreased 20' between each observation. With prac¬ 
tice such observations may be taken with even intervals of 10' of 
arc, which is preferable. The average of an equal number of obser¬ 
vations on the upper and lower limbs eliminates the effect of the 
semi-diameter and gives the mean altitude of the sun’s centre. The 
mean altitude, suitably corrected for index error and refraction, 
may be considered the altitude for the mean time. Then, as shown 
on p. 88, from the formula 


cos t = - — -f — tan d> tan S 

cos <p cos o ^ 

the true hour-angle (£) may be obtained and compared with the 
mean watch time—as shown below. 

Method C. Using Transit, at Any Time. 

This method is almost identical with Method B. If the vertical 
circle is complete, half of the observations should be taken with the 
instrument reversed. This will eliminate the index error of the 
vertical circle, collimation error, and difference of height of the tele- 



108 


scope-standards. After obtMning the mean altitude and mean time 
the form of reduction is identical with that of Method B. Of course 
the sextant is more accurate than an ordinary engineer’s transit, 
chiefly on account of the finer unit readings and the probable error 
of the ievel of the transit. 


Form of Notes, 
(method b.) 


Time. 

Double Alt. 


8 h. 28 m. 03 s. 

29 10 

30 16 

31 23 

32 31 

41° 40' 00" 

42 00 00 

42 20 00 

42 40 00 

43 00 00 


8 33 56 

35 05 

86 13 

37 19 

38 24 

42 20 00 

42 40 00 

43 00 00 

43 20 00 

43 40 00 


8 h. 33 m. 14 s. 

42® 40' 00" 























109 


Longitude. 

If a telegraph-office is accessible, so that “standard time” may be 
obtained electrically, and the watch is compared within 24 hours of 
the time of taking the observations for time, the longitude may - 
probably be obtained to within 1' of arc. The method is most easily 
explained by the solution given below. 


Oct. 27, 1898 
<j) = 39° 55' 51" 

8 (Gr. mean noon) — 12° 53' 47" 

1.55 X 50.64 = — 1 18 

£ _ _ j^o 05 " 

Mean double altitude 42° 40' 00" 
Index error 1 ■ 45 

42° 38' 15" 

Corrected altitude = 21 19 07 

Refraction = — 2 27 

h = 21° 16' 40" 


cos = 9.884693 
cos 8 = 9.988867 
9.873560 
.48553 

_ ( _ .10199 ) 
cos t — .07752 


sin h = 9.559775 

. 9.873560 

. 9.686215 

tan $ = 9.922749 
tan 8 = 9.360522 

. 9.283271" 

t = 47° 21' 0" 
= 3 h. 9 m. 


24.0 s. 


Eq. of time 16 rn. 2.74s. 

,3.16 X -214 = — 0-68 

16 m. 2.06 s. 

Mean local time of appar. noon = 
Hour-angle from meridian 
Mean local time of observation = 
“ watch time of observation = 
Watch slow (mean local time) 

“ k ‘ (75° time) 

Difference of time 
“ “ long. 

Long, of place 


11 h. 43 m. 
3 09 

8 h. 34 m. 
8 33 

1 
2 

0 m. 
12' 30" 
75° 12' 30" 


57.9 s. 
24.0 

33.9 s. 

14.0 

19.9 
10 

50 s. 





















110 


— 42 — 

CONSTANTS OF A LEVEL OF PRECISION. 

Equipment: Level of precision, leveling-rod (or paper scale—see 
next paragraph), and steel tape. 

Location: Ordinarily such tests are made in the field, the only 
requirement being a fairly smooth stretch of 300 to 400 feet; but in 
unfavorable weather a very good test may be made indoors by sight¬ 
ing at an illuminated scale graduated to 50ths of an inch (or to mil¬ 
limeters) from a distance of about 35 feet. at 35' subtends about 
10 seconds of arc, and the 50ths may be easily subdivided by estima- 
tion. The following Notes are taken from an actual in-door test. 

Fixed Constants. 

1. Angular value of one division of bubble-tube . Sight at rod (or 

scale) when the bubble is as near one end of the tube as possible_ 

so that the bubble-reading at each end may be taken. By means of 
the micrometer-screw under one wye, run the bubble to the other 
end and again read the scale (reading the middle cross-wire only) and 
both ends of the bubble. Take six such readings: five differences 
may be obtained from them. Note for each observation the position 
of the center of the bubble; then for each pair the motion of the 
bubble-center and the difference of scale-readings; then for each pair 
the difference of scale per bubble-division. Call r the mean differ¬ 
ence of scale-readings per bubble-division; d = distance from instru¬ 
ment to scale; and v the value of one bubble-division in seconds of 
arc; then 

__ r 
V ~ d sin 1" * 

2. Inequality of the pivot-rings. [The theory and method have 
been developed in Prob. 32 and are not repeated here, especially as a 
careful determination of the value is sufficient for one problem.] 

3. Angular value of the wire interval. Read the extreme cross¬ 
wires on the rod (or scale). Move the telescope by means of the 
micrometer-screw just enough to give different (and independent) 
scale-readings. Take ten such readings. If d = distance from center 




Ill 


of instrument to rod; s = mean difference of wire-readings; (/ -f- c ) 
= tlie distance from the center of the instrument to a point as far 
in front of the object-glass as the cross-wires are behind the object- 
glass; then r, the required quantity, is the fixed ratio of (d — / — c) 
to 5 . When this test is made indoors the distances should be 
measured with corresponding accuracy. 

Daily Tests; 

4. Error of the line of collimation. Read the three wires on the 
rod or scale. Rotate the telescope 180° in the wyes and again read 
the three wires. Take three pairs of readings, moving the telescope 
slightly with the micrometer-screw between each pair. Raise the 
clips (if any) from the collars, and remove the striding-level so that 
the telescope may be rotated with as little jar as possible. The 
difference in the mean wire-readings is twice the collimation error 
for that distance. Reduce the collimation error to its value in deci¬ 
mals of a foot (or meter) per 100 feet (or 100 meters). 

5. Error of striding-level. Observe the bubble-readings at both 
ends when the striding-level is direct and when it is reversed. 
Take three readings with level direct and three with level reversed. 
One half the motion of the bubble-center during reversal indicates 
the bubble inclination. 




112 


FORM OF NOTES. 




Bubble. 



Test. 

Scale. 



Center of 

Motion of 



Bubble. 

Bubble-cen. 



E 

O 




640.5 

+> 1.5 

-34.5 

-16.5 i 

34.7 

38.2 

43.5 

41.8 

38.6 


055.8 

+36.3 

0.0 

+18.2 1 \ 

1 

639.0 

— 2.0 

-38.0 

-20.0 f | 


656.1 

+37.0 

+10.0 

+23.5 1 f 


640.0 

0.0 

-36.5 

-18.3 f | 


656.5 

+38.5 

+ 2.0 

+20.3 ( 

3 



| 


Wire Read. 

Wire Interv’l 




331.0\ 

525.0 / 

194.0 




342.5\ 
536.0/ 

193.5 




335.9\ 
530.0/ 

194.1 




346.8\ 
540.5/ 

193.7 




380.3\ 
574.2/ 

193.9 


O 


394.0\ 
687.5/ 

193.5 




421.8\ 
615.5/ 

193.7 




453.0\ 
646.8/ 

193.8 




471.5\ 
665.5/ 

194.0 




491.5\ 
685.0/ 

193.5 



Diff. of 
Scale- 
readings. 



Diff. of 
Scale 

per Bubble- 
division. 


.441 

.440 

.39.'! 

.385 

.428 



Wire Read., 
Tel. Dir. 

Wire Read., 
Tel. Rev. 

Mean Wire, 
Tel. Dir. 

Mean Wire, 
Tel. Inv. 

i Diff. for 
Inversion. 


( 753.8 

752.0) 





■{ 656.0 • 

654.5 V 

656.17 

654.33 

0.92 


( 558.7 

556.5 | 





(771.2 

769.0) 




4 

1 673.8 

671.1 V 

673.73 

671.53 

1.10 


( 576 2 

574.5| 





786.8 

785.2) 





■< 689.5 

688.0 V 

689.37 

687.40 

0.985 


(591.8 

589.0 ) 





Bevel Dir. 

Level Rev. 









Cen. of Bub., 

Cen.of Bub. 

^Motion for 


E 

O 

E 

0 

Level Dir. 

Level Rev. 

/ 

Reversion. 


18.0 

18.0 

18.5 

17.5 

0.0 

+ 0.5 

+0 25 

5 

22.0 

14.0 

22.5 

13.5 

+4.0 

+ 4.5 

+0.25 


10.5 

25.0 

10.5 

25.0 

-7.25 

-7.25 

0.0 


V 







































































113 



- 

d = 33.86 feet = 406.32 inches 

4174 

r = —— inch = .008348 inch 

50 

r 

V ~ d sin 1" 

•008348 _ 

“ 406.32 X .00000485 

[Prob. 32. Test not repeated here.] 


d = 33.50 feet = 402.0 inches 
/ = 14.4 inches 
c = 7.2 inches 
f c = 21.6 inches 


s 


193.77 

50 


3.8754 


d — f — c 


s 

380.4 

3.8754 


= 98.1 


I 


Mean error in collimation = = .0200 inch at distance of 406 inches 

50 

=? .0049 feet per 100 feet, 
or 4.9 mm. per 100 meters. 

with telescope direct, the line of collimation points downward (the scale 
was inverted) by 4.9 mm. per 100 m. 


Mean inclination of bubble = 0.17 div. =0.72 seconds of arc. 

.•. when level is direct and in center, telescope points doivnward an angle 
of 0.72" or .00035 foot per 100 feet. 






















114 


APPENDIX A. 


PROBABLE ERROR—FORMULAE AND USE. 

[Nothing but a mere statement of working formulae, with their 
use, is here given, which the student must take on trust if he has not 
studied the theory. The proof of these formulae is taken up in 
Geodesy.] 

1. When a number of separate observations of any kind have been 
made with equal care, their arithmetical mean is the most probable 
value of the quantity observed. 

2. The probable error is such a quantity that it would be an eveu 
wager that it is greater or that it is less than the real error. For 
example, if the measurement of a base-line is given as 622.456 ± 
0.096, it means that the arithmetical mean of several equally good 
measurements was 622.456; that the probable error (computed as 
below) is 0.096; that according to- the mathematical laws of the 
theory of probability there is as much chance that the real error of 
622.456 is less than 0.096 as that it is more. 

3. Let n = the number of observations; 

d = the difference between any one observation and the 
arithmetical mean (or the “ weighted mean ”); 

Ei = the probable error of a single observation ; 

E m = the probable error of the mean; 
c = 0.6745, a constant determined by computation accord- 
ing to the theory; 

S = a symbol signifying that a summation should be made 
of all the quantities represented by the following 
letter or letters. 




Then 





115 


For example: 


Angle. 

d 


66° 54' 12".5 
13 .5 

11 .3 
16 .5 

12 .3 
15 .5 

4-1.1 
+ 0.1 
+ 2.3 

- 2.9 
+ 1.3 

- 1 9 

1.21 

.01 

5.29 

8.41 

1.69 

3.61 

Mean, 

66° 54' 13".6 


20.22 


E t = .6745 = ± l /y .33. 

^ = - 6745 '/ 6 |5l = ±0 "- M - 

Angle = 66° 54' 13". 6 ± 0".54. 


4. Weighted Observations. When observations have not been 
made with equal care or are not equally reliable they are “weighted ” 
according to their reliability. If M represents any observation and 

S(wM) 


io its weight, then the “ weighted mean ” = 


2(w) 


observation from the weighted mean to obtain d. 
other symbols as before, 


Subtract each 
Then, with the 


/Stood?) 

® = +¥^T 

JT - , / 

m - c y (2w)(n - 1) 
For example: 


_ j the probable error of an observation of 
\ weight unity; 

= the probable error of the weighted mean. 


Angle. 

IV 

d 


wd 2 

116° 43' 48".81 

5 

+ 0.83 

0.69 

3.45 

48 .76 

4 

+ 0.88 

0.77 

3.08 

49 .53 

5 

+ 0.11 

0.01 

0.05 

51 .56 

3 

- 1.92 

3.69 

11.07 

50 .38 

2 

- 0.74 

0.55 

1.10 

49 .84 

5 

- 0.20 

0.04 

0.20 

116° 43' 49".64 




18.95 


Ey =0.6745 = 1 " 32 - 

Angle = 116° 43' 49".64 ± 0".27. 






























116 


APPENDIX B. 


AZIMUTH. 

The azimuth of a line is the angle the line makes with a true me¬ 
ridian which intersects the line. It is usually reckoned from the 
south and measured to the right. Thus 


an azimuth of 50° means S 50° W 

“ “ “ 160° “ N 20° W 

“ “ “ 255° “ N 75° E 

“ “ “ 350° “ S 10° E 


The ordinary magnetic needle is incapable of very close work even 
when it may be positively known that there is no local attraction and 
when the complicated effects of the various periodic changes in the 
declination are duly allowed for. The recent enormous extensions 
of trolley-lines, telegraph, telephone, and electric-light wires, and 
even barbed-wire fences, have rendered the employment of a magnetic 
needle worse than useless, in many places, as a means of determining 
true azimuth. Even when there is no apparent reason for local at¬ 
traction a needle will often exhibit inexplicable vagaries, as is readily 
seen by making a traverse of a considerable area, measuring all angles 
exactly with the horizontal plates and taking the needle readings of 
all lines. The forward and backward readings of any one line will 
frequently show such discrepancies as to absolutely preclude any at¬ 
tempt at accuracy based on the needle readings. The United States 
Government requires that all surveys of the public lands shall be 
based on azimuths obtained by solar observations. The determination 
of true azimuth should be a part of all property surveys, topograph¬ 
ical surveys, and it is a useful feature (as a check) on extended rail¬ 
road surveys. 

Solar azimuth may be obtained in two general ways : (a) by 
direct observation on the sun with the telescope of an ordinary 
“complete” transit, provided with a colored glass shade (as dem¬ 
onstrated in Prob. 36), and ( b ) by the use of a solar “ attachment ” or 




117 


a “ solar compass” (asdemonstrated in Prob. 35). The first method 
requires very little in the way of special equipment, but necessitates 
the solution of a somewhat tedious calculation for each value ob¬ 
tained. The second method gives the true meridian directly, but re¬ 
quires the use of a somewhat expensive instrument. Accurate work 
requires that the numerous adjustments of both the transit and the 
attachment shall be perfect. An error of adjustment will frequently 
cause an error of azimuth which is several times the angular value 
of the error of adjustment. A proper appreciation of either method 
requires an understanding of certain astronomical relations. 

The figure represents the orthographic projection of the celestial 
sphere, projected on the plane of the meridian of the observer. 

EPZE represents the meridian of the observer. 

Z is the zeuith. 

CP is the polar axis of the earth. 

CE is the trace of the plane of the equator. 

S is the position of the sun. 



EZ = <p = the latitude of the observer. 

ZP = co -(p = 90° - <p. 

80 = h = the true altitude of the sun. 

SZ = co-li = 90° - h. 

Z = the zenith-angle PZS. 

g'p _ s = the declination of the sun north or south of the equator. 
SP = co-5 = 90°- 5. The essential sign of 8 must be considered. 

If the sun is south of the equator (as it is from about Sept. 21 to 
March 21), 8 is negative; and if the declination is (say) S. 20,5= — 
20°. Then co-5 = 90° - 5 = 90° - (- 20°) = 110°. 





118 


From spherical trignometry we have in the spherical triangle SZP: 

, /sin (a - co -h) sin ( s - co-0) 

SID — A/ ' j ; ~r > • • • • 

r SID CO-A SID CO-0 

in which s = \\co-h -|- co -0 -f- co-<5]. 

If it is desired to find the altitude which the sun will have at any 
given “ hour-angle ” (tf) from the meridian, it may be found from 
the formula 

sin h = cos 0 cos 8 cos t + sin 0 sin 8. . . . (2) 

This formula may readily be transformed to give the time when , 
the sun will have a given altitude— 

sin h — sin <p sin 8 


cos t = 


cos 0 cos 8 
sin h 


cos 0 cos 8 


— tan 0 tan 8. 


(3) 


The sun describes each day a path which is approximately parallel 
with the equator, the change in decliuation being very small during 
June and December and fastest when the sun is crossing the equator 
in March and September. The declination of the sun must be 
known for the time of the observation; this is obtainable from the 
Nautical Almanac or Ephemeris as follows: 

To find the declination (5) for any day and hour. Suppose the 
time is 10 a m. Jan. 12, 1897; the place, Philadelphia, Pa. The 
Ephemeris for 1897 gives the •‘apparent declination ” for Greenwich 
at noon Jan. 12, 1897, S. 21° 83' 21' .8 = S. 21° 33'.36 ; difference for 
1 hour = -j- 25". 14 = -f 0'.42. Greenwich is on the prime meridian 
—longitude is 0°. The longitude of Philadelphia (Univ. Pa.) is 75° 
IP. The difference of time is, therefore, 5 hours (to a fraction of a 
minute), and therefore in this case and for this purpose the usual 
“standard 75th meridian time” may be used instead of “ mean 
local time.” In localities which are midway between the standard 
hour meridians the use of standard time instead of mean local time 
would cause a maximurn error during the equinoctial periods of 
29”.5. Therefore standard time may usually be employed for this 
purpose without any error which may be appreciable with an engi¬ 
neer’s transit. 12 m. at Greenwich is 7 A m. at Philadelphia. There¬ 
fore 8 at Philadelphia at 10 a.m. = — 21° 33'.36 + (3 X O'.42) = — 







119 


21“ 32 .10; for 11 a.m. 8 = - 21° 32'. 10 + 0\42 = - 21° 31\68. In 
applying these declinations in the above formula they need only be 
taken to the nearest tenth of a minute of arc. 

Refraction. Refraction causes the suu to appear higher than it 
actually is. Therefore when the altitude of the sun (or a star) is ob¬ 
served, the computed refraction should be subtracted from the appa¬ 
rent altitude to obtain the true altitude. The amount of the refrac¬ 
tion is a function of the temperature and of the barometric pressure. 
For such work as may be done with an ordinary transit or sextant 
the values given in Table VIII will suffice. 

When working with an ordinary transit and at ordinary tempera¬ 
ture and pressure, “ mean refractions,” taken directly from Table 
VIII, may be used uncorrected, for it will usually be found that the 
correction is far within the lowest unit of angular measure. For ex¬ 
ample, at a temperature of 32° F. the refractions should be increased 
about 3%, but a barometric pressure of 28".6 vrould diminish the re¬ 
fractions by about the same amount, and the two conditions together 
would give the same results as in the table. A low atmospheric 
pressure and high temperature will combine to reduce the refraction, 
but a correction of even 10# of the mean refractions can hardly be. 
measured with an ordinary transit when the altitude is over 10°. 

On account of the uncertainties in the refraction at low altitudes, 
it is generally undesirable to take observations at less altitudes than 
10°. 

Effect of Refraction on Declination. The effect of refraction 
requires a modification of the declination-angle which is set off when 
using a solar attachment. When the sun is in the meridian, the dec¬ 
lination is affected by the full value of the refraction; at other posi¬ 
tions the effect on declination equals the true refraction for that alti¬ 
tude times the cosine of the angle at 8 (see the figure on p. 91). The 
values of these effects of refraction for various latitudes, declina¬ 
tions, and hour-angles are given in Table IX. As an illustration : 
latitude = 39° 57'; 8 = + 0° 10'; hour angle = 3.5; take the value 
for latitude = 40°; by interpolation we obtain T 23" = 1.4. The 
effect of refraction being to cause the sun to appear higher, the effect 
(in the northern hemisphere above lat. 23° 27') is to make the appar¬ 
ent position further north. Therefore this correction should oe 
added algebraically to the declination. Therefore in the above case 
the modified declination set off is 0° 10' -|- 1'.4 = 0° 11'.4. 


120 


All values in Table IX which are below and to the right of the 
heavy lines indicate an altitude of less than 10°, in which case the 
refraction is somewhat uncertain and the observations correspond¬ 
ingly unreliable. 

For intermediate latitudes the true values may be found by inter¬ 
polation. For all cases above the heavy lines (which means an altitude 
greater than 10°) the maximum error caused by such interpolation 
will not exceed 3". When the latitude of the place of observation 
is within a degree or so of one of those given in the table, those 
values may generally be used without material error. Otherwise it 
will be advisable to construct a table (by interpolation) for the given 
latitude. 

Inaccuracies in azimuth-work. The azimuth obtained by 
these methods may he inaccurate for several reasons : 

1. The instrument may be in poor adjustment. As it should be 
assumed that the instrument has been adjusted as perfectly as pos- J 
sible for this work, no discussion of this cause is necessary except 
that the error is determinable, as will be shown. 

2. The latitude, declination, or altitude may have been inaccu¬ 
rately determined for use instrumentally or in the formula. 

3. The latitude, declination, or altitude may have been inaccu¬ 
rately set off (or read), or there may be an index error to the vertical 
circle, or the level-bubbles may be out of adjustment. The effect is 
substantially the same whatever the cause. The nominal angle is 
one thing, the real angle is something different and the resulting i 
computed azimuth is more or less inaccurate. 

In Table X is given the effect on the azimuth in minutes of arc 
of an error of one minute in latitude, declination, or altitude. If it 
is discovered after the observations are made that there is an index 
error to the vertical circle, or that an incorrect declination or latitude 
was used, the observations need not be rejected—assuming that the 
error is only a few minutes of arc. By-means of Table X a cor¬ 
rection may be applied to the azimuth obtained which will-give the 
true azimuth. Great care should be taken that the correction is 
applied with its proper algebraic sign. The following rules should 
be observed in applying the corrections ; 

1. If the latitude is used with a larger value than its real 
value, the angle PZS (see figure on p. 91) will be too large. 



121 


2. If the declination is used with a larger (algebraic) value 
than its real value, the angle PZS will be too small. 

3. If the altitude is measured and used with a larger value 
than its veal value, the computed value for PZS will be too large. 
Of course these rules should also be used vice versa. 

An inspection of Tables IX and X will show that accuracy 
depends largely on the time of day and that better work may be ob¬ 
tained in summer than in winter. Observations for azimuth cannot 
be taken at noon ; errors will be extreme for observations near noon, 
and even one hour from the meridian is too close for accurate work. 
The best times are when the sun is as far from the meridian as pos¬ 
sible and yet not so near the horizon that the refraction is uncertain. 
It will usually be possible to so plan work that observations for 
azimuth may be taken at favorable times without wasting tim<5 or 
interfering with other work. 





APPENDIX C. 


ECCENTRICITY AND ERRORS OF GRADUATION OF 
A GRADUATED CIRCLE. 

The readings of two opposite verniers (or micrometers) will usually 
differ slightly, owing to one or more of three causes. These differ¬ 
ences are frequently less than the least unit of measurement in a well-. 
made engineer’s transit, but micrometer microscopes permit the 
reading of such a small unit of measurement that the absolutely 
unavoidable errors of the finest mechanical construction become 
apparent. The three causes are: (a) the micrometers may not be 
180° apart—measured on a circle whose centre is the centre of the 
vertical axis carrying the telescope and the micrometers; (6) the 
centre of this axis may not coincide with the centre of the graduated 
plate; (c) there is more or less error in the graduations. There is 
still a fourth cause, viz.: inaccuracies of reading due to inexperience 
or “personal equation,” but in the investigation of the errors of 
graduation of a circle such errors are necessarily bound up with the 
errors of graduation. Call A the reading of micrometer A in any 
position and B the corresponding reading 
of micrometer B ± 180°; then B — A — the 
difference, which is usually a variable for 
different parts of the circle. Then B — A 
!l for any position of the micrometers includes 
the combined effect of these three causes. 

Cause a. Let AiB lt AiB if etc., represent 
the various positions of the micrometers, m 
representing the vertical axis carrying the 
telescope and micrometers. As may be 
readily seen from the figure, v is a constant for all parts of the 
circle. Then (B — A) includes a constant v, which may be large or 
small and possibly absolute zero. Micrometers are usually adjustable, 
so that v may be reduced to a very small quantity. 

Cause b. Since v is a constant in ( B — A), we may assume for 
simplicity that it has been eliminated, or that m, the centre of the 






123 


axis supporting the telescope, is on a line joining A and B in every 
position. Let C be the centre of the graduated circle. Then AC A! 
measures that part of (B — A) due to cause b. The effect of cause 
b is plainly seen to be variable for various positions of the microm¬ 
eters, and when B reaches h (or k) the effect is zero, i.e., mh is the 
“line of no eccentricity.” Let h be the 
reading at /t, and t the reading for any posi¬ 
tion of B ; then {t — h) = BCh; mC = e, 
the linear eccentricity; BC = JR; mC -*■ BC 
= e -r- R, which may be called e' sin 1”. 

From the triangle mCB (in any position of 

B) we have sin mBC = sin BCh. Let 
Bin 

x = AC A! = 2mBC. Since mC is exceed¬ 
ingly small, Bm is practically equal to 
BC ; mC -f- Bm = e -i- R = e" sin 1”. .*. sin = \x sin 1" = 

e" sin 1" sin (t — h). x = 2e" sin {t — h). But note that if the tele¬ 
scope is swung around 180° from any given position the algebraic 
sign of x is changed, although x is numerically the same as before. 
Therefore 2e" sin (t — h) is numerically the same but of opposite 
sign for all pairs of positions 180°* apart. 

Cause c. Errors of graduation and reading are variable and 
wholly irregular. Call them g. Evidently the error of any gradua¬ 
tion is its variation from a position in an ideal system whose differ¬ 
ences from the actual marks are as small as possible and are equally 
positive and negative. Therefore the average of all the g’s for a 
large number of graduations, taken at regular intervals around the 
circle, is zero. 

For any position of the micrometers (B — A) — v-\-2e" sin (t — li)-\-g. 
Suppose that readings for (B — A) are systematically taken at poiuts 
of the circle 10° or 20° apart, a set of variable values will be found. 
Since g is assumed to average zero, and since for every value of 
2e" sin (t — h) there is an equal value of opposite sign, the mean of 
all the values of (B — A) is evidently v. Then if v is subtracted 
from each value of (B — A) we have (B — A) — v — x -j- g. But if 
(B — A — v) d ° = x + g, then (B — A - v)d° + iso 0 = — x — g. Let 
[(B - A - v) d ° - (B - A - v) d ° + iso 0 ] = 2D = 2x + 2 g. Then 
D = x _j_ g = 2e" sin (t — h) + g. It seems impracticable to wholly 
eliminate g from the values of 1 ), but it may be seen from the follow- 





124 


ing method of solution that if we call D = 2ef' sin (t — h) and obtain 
values for e" and h, the values obtained will be very nearly correct, 
since the mean value of g is zero and in the summations will almost 
cancel out. We thus have a series of variable numerical values for 
D, each of which equals 2e" sin ( t — h), in which e" is a required 
quantity, t has certain known values varying by 10° or 20 c , and h is 
unknown. The only practicable method of solution is.one involving 
the principles of the Method of Least Squares. The numerical work 
is simple and may be accepted by the student until he has mastered 
the theory. Only the numerical process will be here given. 

For each numerical case we have (assuming that the readings are 
taken at each even 20° around the circumference) 

A = 2e" sin (0° - h), D a = 2e " sin (20° - 7i), etc., 
which may be written 

Zb = 2e" (sin 0° cos h — cos 0° sin h), 

Di = 2e" (sin 20° cos h — cos 20° sin h), etc. 

Multiply Di by sin 0°, Z> 2 by sin 20°, etc., and add the results; then 
multiply Di by cos 0°, D 2 by cos 20°, etc., and add the results. We 
then have 


2(Z) sin t ) = 2e" cos h (sin 2 0° -|- sin 2 20° + .. . + sin 2 160°) 

— 2e" sin h (sin 0° cos 0° + sin 20° cos 20° -f . . . -f sin 160° cos 160°). . (a) 
2(Z) cos t) = 2e" cos h (sin 0° cos 0° + sin 20° cos 20° +. . . + sin 160° cos 160°) 

- 2e" sin h (cos 2 0° + cos 2 20° -f . . . + cos 2 160°).(6) 


In (a), sin 0° cos 0° = 0, sin 20° = sin 160°, and cos 20° = — cos 160°; 
therefore sin 20° cos 20° -f sin 160° cos 160° = 0 ; and similarly all 
the quantities in the second term of (a) and also the first term of ( b) 
will cancel out. It may be readily shown that (sin 2 0° -|- sin 2 20° 

1 180 ° 

+ s i n2 160°) = 4.5 precisely, i.e., — If the uniform 

interval were 10°, the sum would be = 9. Likewise it may be 

A 1U 

shown that (cos 2 0° -f- cos 2 20° + . . . cos 2 160°) = 4.5. Then 

2(D sin t) = Tie" cos h, in which n 
2(D cos t) = — ne" sin h ; 



2{D cos t ) 
2(D sin t) 


— tan 7t. 






125 


h becoming known, we may solve for e" from the equation 

c " — — ^S(-P cos t) 

~~ n sin h _ * 

Knowing the reading h for the line of no eccentricity and the value 
in seconds of the eccentricity ( e "), the variable values of x are obtain¬ 
able. By subtracting each value of x from the corresponding value 
of B — A — v the error of graduation (g) for that angle may be found. 
We know that the values of x must vary according to a definite 
system, depending on the position of the line of no eccentricity, and 
the above method gives the most probable system. The method gives 
the most probable constant value for v, the most probable system of 
values for x, the most probable position of the line of no eccentricity 
and the angular value of the eccentricity, and throws the remainder 
into errors of graduation and reading. The linear eccentricity (e) 
equals e"H sin 1". 

For a numerical illustration, see the solution given in Problem 33. 



APPENDIX D. 


NOTES ON THE MANAGEMENT OF A STUDENTS’ 
TOPOGRAPHICAL-HYDROGRAPHICAL SURVEY. 

The following directions were originally written for the purpose 
of supplementing the general treatment of the subject, as given in a 
text-book on surveying, by specific instructions applied to the locality- 
of the survey—a section of Fairmount Park, including a stretch of 
the Schuylkill River and a considerable area along the banks. They 
have been incorporated, with some revision, into this book because it 
is believed that they are sufficiently applicable to any similar survey 
to warrant their publication. 

The triangulation methods are those for tertiary work, and the 
directions for “precise” base-line work, latitude and longitude, were 
inserted as exercises for the upper classmen, and not because they 
are essential to the proper conduct of the survey. 

A. Reconnoissance ; Location of Stations, 

1. Two parties are necessary—one on each side of the river. 
Each party should be supplied with stakes, a heavy hatchet or axe, a 
pocket sextant (or prismatic compass), three transit-rods with flags 
tacked on, and a light crowbar. 

2. In general four lines will run from each station—two to stations 
above and below on the same shore and two to stations on the oppo¬ 
site shore. The distance between stations along one shore should 
be, when possible, a little more than the width of the river so as to 
obtain nearly equilateral triangles. 

3. The forward rodman on each side is controlled by the party on 
the other side. When the forward rodman reaches a suitable loca¬ 
tion that gives satisfactory angles (as near 60° as possible) to both 
parties he remains there until dismissed by both parties. The rear 
flagman on each side occupies the station last occupied by the main 
party. 

4. The angles between stations should be measured with a pocket 



127 


sextant to the nearest degree. These should be plotted at night so 
that any necessary modifications can be promptly made, and then the 
stations can be properly numbered as a basis for future field orders. 

5. All triangulation stations should be marked by stakes 2" X 2" 
X 18", centred with tacks, driven to within 4" of the ground, using 
a crowbar to make a hole if necessary. The stakes for the ends of 
the base-line should be longer ; they should project at least 12". 

B. Base-line. Precise Method ; accuracy 1 : 1000 000. 

1 . The site for the base-line should be nearly level, and on as uni¬ 
form a surface as possible. The roadways on the river-banks gen¬ 
erally afford the best sites. 

2. A hub should be set every 300 feet, measuring from one end ; 
also, a hub should be placed within 10 feet of the other end and at an 
even 10-foot distance from the last 300-foot hub placed. The hubs 
should be long enough to project 12". At every 50 feet between 
the hubs 3-foot stakes should be driven so that the base-line would 
pass about ±" from the face of the stake. Screw-eyes with 3" hooks 
should be screwed into the faces of these stakes so that the bottom 
of each hook is on the grade-line between the tops of the adjacent 
hubs. The hooks should have the points bent upward, and the 
screws should be so set that the hooks will point outward and swing 
freely in both directions. Strips of zinc should be nailed to the tops 
of the hubs. While obtaining the grade-line between hubs, the 
differences of elevation of all adjacent hubs should be noted. 

3. The tension-frames* should be used with a tensiou of 16 lbs. 
The rear end of the tape is placed exactly on the station mark, the 
forward end screwed up until the tension of 16 lbs. is obtained, 
every hook support critically examined to see that the tape rests with 
perfect freedom in the hooks, and then a fine mark is scratched in 
the zinc strip on the hub with a fine steel point. Then the tape is 
moved ahead. While measuring, two thermometers should be tied 
to the tape near the 100-ft. and 200-ft. points (marked 90 and 190 on 
the U. Pa. 300-foot tape), and the readings taken to the nearest 


* The tension-frames referred to were designed by the author. They pro¬ 
vide for accurate adjustment of the tape over the hubs and an extremely 
accurate measurement of the tension—by a scale-beam and not by a spring. 
The frames are of tubing, with pointed ends, and are easily adjustable, tripod 
fashion, over the hubs. 





128 


tenth, of a degree at every measurement. These thermometers 
should be carefully standardized and their exact errors known. 
The last sub-distance (less than 10 feet) is measured with a fine 10- 
foot steel tape, graduated to .01' and estimated to .001'. 

4. Measurements should not be made, unless for mere practice, 
while the sun is shining on the tape. A calm, cloudy day should be 
chosen, and even better results are obtained during a drizzling rain. 
In default of cloudy days, measurements should be taken immedi¬ 
ately after sundown or before sunrise. For the best results the 
earth and atmosphere should have the same temperature. This 
probably occurs during the evening, but then lanterns must be used, 
endangering the accuracy.' 

C. Base-line. Ordinary Method; probable accuracy, 1:50000. 

1. Measure the base-line three times with an ordinary 100-foot steel 
tape, using fine brass plumb-bobs to plumb down the tape ends to 
the tops of stakes set every 100 feet- The mean temperature of the 
air should be noted with a pocket thermometer, and corrections 
made accordingly. Care should be used to have a uniform tension 
of as near 16 lbs. as possible. A spring-balance should be used for 
this. 

2. The mean of these measurements may be used as the final 
value for the triangulation should anything prevent the “ precise ” 
measurement (B). 

3. A check-base should be measured at or near the other end of 
the system of triangulation. 

D. Azimuth. 

1. Azimuth may be determined by a set of observations with a 
solar attachment. At least twenty observations should be obtained 
having an extreme range not greater than 5' of arc (see Prob. 35). 
The observations should be taken at a triangulatiou station, and each 
observation consists in finding the angle between the meridian, as 
determined by that observation, and a line to another triangulation 
station. 

2. Azimuth may also be determined by one of the methods given 
in Problem 36. Some such method is necessary when the equipment 
does notinclude a solar attachment. 


129 


3. A valuable check on the work may be obtained by taking azi¬ 
muth at stations near each end of the triangulation, making correc¬ 
tions, if necessary, for the convergence of meridians. 

4. True latitude (as close as the reading limit of the vertical arc of 
the transit) should be known. If it is not obtainable from an accurate 
map of the section of the country, it must be observed as provided 
below. 


E. Latitude. 

1. Observations for latitude may be made by observing the culmi¬ 
nation of the sun at noon by one of the methods described in Problem 
37 or in Problem 40. 

2. Observations for latitude should be taken at two triangulation 
stations which differ considerably in latitude. These observations 
may be checked by allowing 101 feet for each second of arc in the 
difference of the latitudes obtained and comparing this result with 
that computed directly from the triangulation. 

F. Longitude. 

1. Observations for longitude may be made by the method devel¬ 
oped in Problem 41. Azimuth and Latitude should have been 
previously determined at the same station. Unless it is possible to 
obtain standard time from a telegraph-office within 24 hours of the 
time of taking these observations, it will be useless to attempt the 
work unless a well-rated timepiece is obtainable. 

G. Triangulation. 

1. Sightings may be taken (1) at a transit-rod, held by a rodman. 
pointing as near the bottom of the rod as possible, noting by the* 
vertical cross-hair the verticality of the rod, or (2) at a transit that 
may be occupying the station, pointing at the plumb-bob or at the 
vertical spindle underneath the plates. When it is evident that a 
station occupied by a transit party is being sighted at by another 
party, care must be taken to facilitate their work by not obstructing 
the line of view and always giving them a good mark to sight at. 
If a transit which is being sighted at is removed from the station, 
it should be promptly replaced by a rod. 

2. Angles should be measured by the method outlined in Prob. 


130 


13. Measure by repetition until six consecutive values are obtained 
having an extreme range not exceeding 2' of arc. Compute the 
mean value. 

3. All triangles must “ close ” with an error not exceeding 1' of 
arc. 

4. As fast as the angles of the triangles are satisfactorily meas¬ 
ured, the sides should be computed from the measured base, the 
true azimuth of each triangulation line determined, and the whole 
triaugulation plotted. 


H. Levelling. 

1. A line of levels should be run along each bank of the river, ' 
noting the elevation of each triangulation station and also of the 
water surface near each station. 

2. A tide-gauge should be set up in a suitable place, the elevation 
of its zero determined, and readings to the nearest tenth of a foot 
taken as often as proves necessary, while the line of levels is run and 
while stadia topography is being taken. 

I. Soundings. 

1. Soundings to a depth of 18 to 20 feet may be made with a 
sounding-pole. A sounding-line with weight wijl be needed for a 
large pg,rt of the soundings above the Girard Ave. Bridge (max. 
depth about 36 feet). 

2. Lines of soundings should be run nearly perpendicular to the 
course of the river and about 300 to 500 feet apart; the individual 
soundings about 50 feet apart. 

3. The soundings should be located by two transits on shore 
pointiugat each other and at the boat. As a check, a sextant in the 
boat points at the two transits. Each transit is set at 0° on the other 
transit, the lower plate always clamped, the upper plate free, the 
movement of the boat constantly followed with the telescope. 

4. At the instant of taking a sounding, a signal will be given 
from the boat—a waving flag or a whistle. The shore parties will 
note first the time of the signal, and second the angle to the boat. 
Since it will not increase the accuracy of the plotting, the angles 
need not be read closer than the nearest 10' or 15', and so it will be 
easier and better to take the reading directly from the index, not 


131 


even reading the vernier. The transitmen should point at the sound¬ 
ing-pole as nearly as possible. 

5. The shore parties and the boat party must be supplied with 
watches, previously compared to a close fraction of a minute. It 
will save trouble in identifying the shore and boat records if the 
soundings (and signals) are made always at an even minute of time, 
and never oftener than once per minute. An occasional signal of 
two whistles (marked as such in all the note-books) will also aid in 
identifying the soundings, especially when mistakes have been made. 

6. The shore parties should be located at triangulation stations 
when they are suitable. Otherwise, points which are especially 
desirable for this purpose may be selected and their locations accu¬ 
rately determined from the triangulation stations. 

7. The following form of notes should be used for the note-book 
in the sounding-boat: 


Line. 

Time. 

Sextant 

Angle. 

Angle. 

Depth. 

Bottom. 

4—X> 2 

2:34 

E —boat —D 

62° 20' 

15.7 

Sandy 


2:36 


68° 40' 

18.2 


The right hand page should record the personnel of the party and 
any other desired information. Z) 2 means the second point on the 
shore below Sta. D toward (or away from) which a line of sound- 
ings has been run. The note-books of ihe shore parties should 
observe the following form: 


Inst, at 

Line. 

Time. 

shore 

Angle. 

Angle 


A D 

4—D 2 

2:34 

2.36 

boat— D—E 

45°10' 

42° 30' 



8. During the evening following the sounding work, the sound¬ 
ings in the sounding-book should be numbered consecutively and 
the corresponding shore angles numbered with the same numbers, 
the correspondence being determined by the identity of time, veri¬ 
fied by double signals, etc. These checks of double signals, etc., 
are often proved valuable. 






























J. Topography. 


1. Topography is taken by stadia lines, which must always begin 
and end on triangulation stations or other points equally well 
determined. Beginning at some station, the telescope should be 
pointed at some other known station, with the plates set at the known 
azimuth of that line. 

2. Distance and vertical angle are read both for fore-sight and 
back-sight between stadia stations. Side shots are taken to all 
desired points within a radius of 500 to 600 feet. 

3. One line should be run near enough to the shore line to take 

in all the configurations of the shore and also a belt of land from 
200 to 600 feet wide—varying with the interference of trees and 
bluffs. The vertical angles to shore points should be noted with 
especial care, as it furnishes a valuable check on the levels. , 

4. The course of the other meander lines should depend largely 
on the topography. In general, they should follow roads, etc., so 
as to pass as near as possible to the important points that must be 
observed, and yet must pass near enough to the other lines that no 
feature of topography will be omitted. 

5. If the transits contain fixed stadia wires, a careful test of them 
should be made at the beginning of the season’s work, and if the 
interval does not correspond with the graduation of the rod, a 
reduction coefficient should be computed and applied to each read¬ 
ing. If the wires are adjustable, tests should be made frequently, 
even daily, considering how simple it is.. Measure off a base-line 
600 feet long on level ground near headquarters. Each morning, 
before beginning stadia work, set up the transit at a distance 
“f-\- c” behind one end of the base-line, and read the rod held at 
the other end. An error of more than 1 : 600 should be immedi¬ 
ately adjusted, after consultation, with an instructor. If the first 
“set-up” of the transit is to be at a triangulation station, it may be 
more convenient to begin there at once and point to another station 
whose distance is known, and take the reading, making due allow¬ 
ance for the “/ + c.” 

6. Further instructions on this subject, together with a form of 
notes, may be found in the instructions for a preliminary survey 
(railroad work) on pp. 110-113. 


K. Computations ; Plotting. 


1. The closure of triangles within the required limits must be de¬ 
termined as soon as possible, and the angles remeasured if necessary. 

2. The azimuth and length of all triangulation lines must be com¬ 
puted as soon as sufficient data are obtained. 

3. To check topographic work, the co-ordinates of all triangulation 
stations from some chosen meridian and parallel should be com¬ 
puted before topographic work commences. 

4. The latitudes and departures of the lines between stations of 
the topographic work should be computed and checked with the 
previously computed co-ordinates of the triangulation stations to 
which the stadia lines run. 

5. For such lines as are run on this survey—probably less than £ 
mile between such stations as are used to check on the errors should 
be within 10 feet horizontal distance and 1 foot in vertical height.* 
Greater errors than these call for a thorough revision of notes to 
discover a possible error, and perhaps a resurvey. 

6. The triangulation should be plotted to a scale of 200 feet to 
the inch as soon as the computations are complete. 

7. All soundings should be plotted duriug the progress of the 
survey, or at least sufficient of them to show the general accuracy 
and reliability of the notes. 

8. All topographical notes should be plotted during the survey 
(at least in pencil)—plotting first the transit stations until the line 
checks with sufficient accuracy on some known station and then 
plotting the side shots, locating all drives, walks, buildings, rail 
roads, bridges, etc. 


* ThPCP errors are verv large considering what accuracy is possible with the 

closer work than this can be expected of 

inexperienced students. 





APPENDIX E. 


NOTES ON THE MANAGEMENT OF A STUDENTS’ RAIL¬ 
ROAD SURVEY. 

It should be remembered that students’surveys are made primarily 
to teach the students. Therefore, although it is desirable to simulate 
regular professional practice as far as possible, matters of speed and 
economy should sometimes be subordinated, so that the students 
may receive the maximum benefit. 

Reconnoissance. 


A reconnoissance survey is a very hasty examination of a wide 
belt of country to determine which routes will best justify further 
and closer instrumental examination. In practice they vary from a 
trip made on horseback, depending principally on “ judgment ” 
and “an eye for country,” to an instrumental survey differing but. 
little in character from a preliminary survey. The prime object is to 
determine with the greatest rapidity and least expense the salient 




features which will condemn one proposed route and commend an- 

lines 


other, or which will narrow down the choice to two or three 
that are so evenly balanced that a closer survey of each is necessary 
to decide between them. In places where the topographical maps 


of the U. S. Geological Survey (or other similar maps) are avail¬ 
able, no reconnoissance (at least so far as th q physical characteristics 
of the country are concerned) is necessary ; a general route may be 
chosen from the map and the preliminary survey at once made. In 
all civilized countries some sort of a map can generally be found. 




135 


although coarse and inaccurate. The method here outlined presup¬ 
poses the availability of a map which will give the relative horizontal 
positions of prominent features of the country, such as cross-roads, 
streams, etc., with such a degree of accuracy as county maps, as or¬ 
dinarily made, are found to have. In the absence of such a map a 
better method of obtaining distances should be adopted. 

The instrumental equipment consists of a mercurial barometer, an 
aneroid, a pocket thermometer, a pocket compass, and a haud- 
level. The mercurial barometer should be maintained at headquar¬ 
ters and read every half-hour during the periods when the aneroid is 
being read in the field. The altitude above sea-level of each loca¬ 
tion for headquarters should be obtained if possible by a compari- 
some with some known altitude, points on another railroad, a geo¬ 
detic survey station, etc. If nothing of the kind is available, extra¬ 
ordinary care should be used in obtaining the difference of elevation 
of the different locations for headquarters. 

Distances may be estimated, paced (aided perhaps by a pedometer), 
or measured when convenient with an odometer. Azimuth may be 
measured with a pocket compass, and altitudes with the aneroid. 

The survey line should start from some known place on the map 
(generally some town), and should pass through all points which will 
probably be determining points on the road. In the course of a few 
miles it will probably pass within range of some easily identified 
point—a road crossing, bridge, hamlet, etc. “Side shots ” should 
be taken to all such points and to any points whose position might 
have some bearing on the proper location of the road. Sketches 
should be made on the right-hand pages of the note-book on a scale 
of 400 feet to the inch, sketching not only the topographical features, 
but also th eform of the contours. During the evenings following 
each day’s work in reconnoissance the aneroid observations should 
be reduced (by comparison with the plotted mercurial readings for 
the day) and entered in the note-book. (See Prob. 22.) Then the line 
should be plotted on the map of the country that has been obtained 
for a basis, modifying the location of the points by making the plot¬ 
ting of well-defined points agree with the (probably) more correct 
map used as a basis and drawing in contours as well as the data will 
permit, utilizing the aneroid elevations. From this rough map may 
be decided what line (or lines) will justify a further and closer ex¬ 
amination by means of the “preliminary survey.” 


136 


Form of Notes—Reconnoissance Survey. 


Observer 

at 

Bearing. 

Distance. 

Aneroid. 

Reduced 

Elevation. 

Sighting at 

A 23 



29" .625) 

63° V 
10:45 \ 

** 


S 70° W 

500' 


hill about 
90' high 




N 5° E 

600' 



road corner 




29". 922) 

64° > 

** 

(creek 
-( under 




10:20 ) 


(bridge 

A 22 

N 5° E 

N 25° W 

50' 

950' 

29".850) 

66° y 
10:10 j 

** 

bridge 
a 23 

• 


• 





** Computed and entered after the return to headquarters. 


Preliminary Survey. 

The party should consist of a transitman, recorder, two rodmen 
with stadia-rods, stakeman, leveller, and level-rod man, and a topog¬ 
rapher.* 

Azimuth at the first station should be determined as accurately as 
possible by a needle-reading, setting vernier A at zero, with the needle 
pointing south. A splendid check on azimuth may be obtained by 
using a solar attachment at the beginning and end of the survey, and 
also at intermediate points at intervals of a few miles, making due 
allowance for the “convergence of meridians.” The extra time re¬ 
quired for these check observations is insignificant. Carry the trav¬ 
erse throughout the survey by the same method as in Prob. 18. 


♦See note on page 138. 




















137 


Read up the page. 



Particular care should be taken in observing the distances be¬ 
tween stations. Not only should the distance be read both during 
fore-sight and back-sight observations, but if the sighting is not very 
clear for any reason repeated sightings should be taken as suggested 
in Prob. 17. If the back-sight does not agree with the previous fore¬ 
sight, the error should be located until there is no doubt of the true 
distance. 

Take “ side shots ” to all fence corners, road bends, river bends, 
buildings, salient points of ground surface, and any other points 
necessary to obtain a topographical map with 5-foot contours, 
throughout the belt of country traversed by the survey. The width 
of country necessary to be surveyed varies with the topography. It 














138 


should include a considerable width each side of any probable loca¬ 
tion of the road in that belt. Even when topographical features de¬ 
termine the location to within a few feet, the topography should be 
taken for 200 feet on each side. From this minimum width it should 
be increased up to 1000 feet each side as the uncertainty of possible 
location increases. Much time and work can be saved by limiting 
the observations that are taken only for contours to points that are 
the salient points—in the bed of a stream or dry gully, the comb of a 
ridge, the edge of a bluff, etc. 

The elevations of all stations should be determined with a wye- 
level to the nearest tenth of a foot. Benches and turning points 
should be read to the nearest hundredth. A bench-mark should be 
established every half-mile. 

A common method of taking topography is to depend entirely on 
free-hand sketches made in the field, the topographer obtaining all 
possible aid from azimuths and stadia readings as they are read and 
called out, and frequently by using a hand-level to observe the 
approximate course of 5-foot contours. A much more satisfactory 
method is to make an accurate drawing of all the necessary topog¬ 
raphy while in the field. A sketch-board 24" X 24" mounted on 
an ordinary tripod is a satisfactory drawing-table. Sheets of What¬ 
man’s drawing-paper having a large (13") protractor engraved on 
them can be used, and all azimuths can be plotted with all necessary 
accuracy by means of a pair of triangles or a parallel ruler. A 
stadia slide-rule is easily carried and may be used to immediately 
furnish all stadia reductions. The advantage of this method lies in 

Form of Notes—Preliminary Survey. 


Inst, at 

Wire 

Int. 

Azim. 

Vert. 

Angle. 

Hor. 

Dist. 

Diff. 

Elev. 

Sighting at 

164.3 







A 26 

423 

265° 18' 

— 0° 16' 


— 2.0 

A 25 


564 

62° 47' 

+ 0* 52' 


+ 8.5 

A 27 


848 

322- 14' 

+ 3° 21' 

845 

— 49.3 

fence cor. near barn 


168 

138° 20' 

-f 18° 40' 

151 

+ 51.0 

edge bluff 


etc. 

etc. 

etc. 


172.75 







A 27 

563 

242° 47' 

— 0° 51' 


- 8.4 

a 26 


470 

78° 22' 

+ 1° 18' 


+ 10.7 

a 28 
















189 


tne mstant discovery of errors while on tne ground, and in the facility 
with which complicated topographical features may be surveyed and 
plotted without any uncertainty as to the interpretation of the notes. 

When a mile or more of this preliminary work has been surveyed, 
plotted, and inked , it may be practicable to begin the “paper loca¬ 
tion,” which may be studied and drawn on the preliminary map. 
From the intersection of this line with the contours an approximate 
profile may be drawn which will be valuable in giving an idea of 
the amount of cut and fill, and will be especially valuable in com¬ 
parison with one or more profiles similarly obtained from other pro¬ 
posed locations. 


LOCATION SURVEY. 

The location party may consist of a transitman, a recorder, two 
tapemen, two rodmen, a stakenmn, two cross-section men, a leveller, 
and a level-rod man. When short of men the transitman may keep 
notes and one rodman may be dispensed with.* 

Stakes should be located at every station (at least) on tangents, 
every 50 feet on curves up to 6 3 , every 25 feet on sharper curves; also at 
all abrupt changes in the profile (both to increase accuracy in chain¬ 
ing and to locate points for the leveller). Stakes should be marked 


* To render these student surveys as free from objection (and expenses for 
damages) as possible, cutting and clearing is avoided wherever possible. With 
a superabundance of men, branches can be held back and cutting often avoided. 
Therefore no reference has been made to axemen, either in this or in the prelimi¬ 
nary survey. The force of men is ample to supply an axeman for all cutting that 
is indispensable. The necessity for clearing is allowed considerable weight in 
selecting the locality for these surveys. 


Eleva¬ 

tion. 

Needle. 





115.0 

215.3 

N 85° 15' E 
S 62° 45' W 



N 62° 40' E 
S 78° 15' W 











140 


on the rear side, with the number of the station and its plus-distance, 
if any. Transit hubs should be driven flush with the ground, the 
exact centre marked with a tack, and should have a witness-stake set 
three feet to the left, with the distance and P.C. or P.T., etc., marked 
on it, the marking facing the hub. 

The leveller should observe the elevation of all stations and sub¬ 
stations, and should even interpolate points with a tape and observe 
their elevations when it will add to the accuracy of the profile. The 
B.M.’s of the preliminary survey should be used and checked as long 
as they are within range of the location line. New B.M.’s may be 
interpolated when necessary. The rod should be read to the near¬ 
est tenth on stations and to the nearest hundredth on turning-points' 
and bench-marks. The profile of each day’s work should be 
promptly plotted at night on profile-paper, Plate A, using horizontal 
scale 200 feet to the inch, vertical scale 20 feet to the inch. 


Form of Notes. Location Survey. 


Sta. 

Alignment. 

Vernier. 

Tang. Defl. 

Calc.Bearing. 

Needle. 

a -1-47 

P.T. 

6° 10' 

12° 20' 

N 1° 24' E 

N 1° 30'E 

52 

+>* 

5° 13^' 




51 


3° 13J4' 




50 

go|" 

o H 

1° 13' 

' 



a +39 

^P.C. 





49 






48 






A +50 

P.T. 

7° 20' 

14° 40' 

N 10° 56' W 

N 11° 0' W 

'47. 

• T* 

o 

6° 40' 




A +50 

46 

^ cd 

sh 11 

6° 00' 

5° 20' 




45 


4° 00' 




A +50 
44 

W*S bfl 
~ a 

EH 

3° 20' 

2° 40' 




43 


1° 20' 




A 42 

P.C. 



N 3° 44' E ! 

N 3° 40' E 












141 


The transitman should “ tie on ” to the stakes of the preliminary 
survey, whenever possible, not only as a check on the work, but also 
to observe and correct any departure from the line as laid out on ac¬ 
count of inaccuracies in the paper location. 

The cross-section men should work by the methods and with the 
form of notes elaborated in Prob. 28. The width of the cross-sections 
may be varied according to the amount of cut or fill that would evi¬ 
dently be required. They should be made about 50 feet wider than 
the probable width of top of cut or base of fill. They should be 
ploited at night on cross-section paper. 

The location map should be plotted at night to a scale of 200 feet 
to the inch, drawing in 5-foot contours from the cross-section notes 
and borrowing other topographical details not observed during the 
location survey from the preliminary survey map. 

The form of notes for the location survey should be as on pp. 114, 
115. 

(Read up the page.) 








142 



When the location profile has been completed for a sufficient dis 
tance so that the “grade-line” may be laid out, it should be drawn 
in, and then a list of “surface elevations” and “grade elevations” 
for each station and substation should be entered in the slope-stakes 
book ; also (as near as possible) the position of grade points for cen¬ 
tre line and edges of road bed. Then, knowing the “centre cut (or 
fill),” the positions of the slope-stakes, breaks, etc., are to be found 
and notes entered as described in Prob. 29. When men are plenty 
the party may consist of a leveller, computer and recorder, and two 
tapemen, one carrying the rod. 

To measure the volume of the terminal pyramid when emerging 
from a cut, take a cross-section at point where the side of the road- ' 
bed that first emerges from the cut actually comes out. Measure the 
perpendicular distance from this cross-section to the point where the 
other side of the road-bed emerges. The area of the cross-section 
times one third of this distance gives the volume. The volume of 
the initial pyramid of fill is similarly obtained. If the width of road¬ 
bed in cut and fill is the same, the altitudes of the pyramids are the 
same, and the apex of one lies in the base of the other ; otherwise 
they are separated by a small distance. If the line of intersection of 
the road-bed and the surface is nearly perpendicular to the line of 
road, no appreciable error is involved in treating the terminal vol¬ 
umes as wedges, having as bases the nearest convenient cross-sec¬ 
tions, and as altitudes the distances measured along the road-bed from 
these cross-sections to the intersection of grade-line with the surface. 

To find grade points, set the target of the level-rod at a reading 
equal to the difference between the height of instrument and the ele¬ 
vation of grade-line at the point where the grade point is supposed 
to be, as determined by the profile. The uncertainty as to grade ele¬ 
vation, on account of the uncertainty as to the exact grade point, is 
usually too small to be considered. Have the rodman move along 
the centre line (or the side lines of the road-bed), and the point where 
the target is level with the instrument is the grade point. 














































































f 







































144 


TABLE IV.—SQUARES OF NUMBERS. 



* * o 

* * 1 

* * 2 

* * 3 

* * 4 

* * 5 

* * 6 

^ 

* * 8 

* * 9 

10 

10 000 

10 201 

10 404 

10609 

10 816 

11 025 

11 236 

11449 

11 664 

11 881 

11 

12 100 

12 321 

12 544 

12 769 

12 996 

13 225 

13 456 

13 689 

13 924 

14 161 

12 

14 400 

14 641 

14 884 

15129 

15 376 

15 625 

15 876 

16129 

16 384 

16 641 

13 

16 900 

17161 

17 424 

17 6S9 

17 956 

18 225 

18 496 

18 769 

19 044 

19 321 

14 

19 600 

19 881 

20164 

20449 

20736 

21 025 

21 316 

21 609 

21 904 

22 201 

15 

22 500 

22 801 

23 104 

23409 

23 716 

24 025 

24 336 

24 649 

24 964 

25 281 

16 

25 600 

25 921 

26 244 

26 569 

26 896 

27 225 

27 556 

27 889 

28 224 

28 561 

1 ? 

28 900 

29 241 

29 584 

29 929 

30 276 

30 625 

30 976 

31329 

31 684 

32 041 

18 

32 400 

32 761 

83 124 

33489 

33 856 

34 225 

34 596 

34 969 

35 344 

35 721 

19 

36100 

36 481 

36 864 

37249 

37636 

38025 

38 416 

38 809 

39 204 

39 601 

20 

40 000 

40 401 

40 804 

41 209 

41 616 

42 025 

42 436 

42 849 

43 264 

43 681 

21 

44 100 

44 521 

44 944 

45 369 

45 796 

46 225 

46 656 

47 089 

47 524 

47 961 

22 

48 400 

48 841 

49 284 

49 729 

50176 

50 625 

51 076 

51 529 

51 984 

52 441 

23 

52 900 

53 361 

53 824 

54 289 

54 756 

55 225 

55 696 

56 169 

56 664 

57121 

24 

57 600 

58 081 

58 564 

59 049 

59536 

60 025 

60 516 

61 009 

61 504 

62 001 

25 

62 500 

63 001 

63 504 

64 009 

64 516 

65 025 

65 536 

66 049 

66 564 

67 081 

26 

67 600 

68 121 

68 644 

69169 

69696 

70 225 

70 756 

71 289 

71 824 

72 361 

27 

72 900 

73 441 

73 984 

74 529 

75 076 

75 625 

76176 

76 729 

77 284 

77 841 

28 

78 400 

78 961 

79 524 

80 089 

80656 

81 225 

81 796 

82 369 

82 944 

83 521 

29 

84 100 

84 681 

85 264 

85 849 

86436 

87 025 

87 616 

88 209 

88 804 

89 401 

30 

90.000 

90 601 

91 204 

91 809 

92 416 

93 025 

93 636 

94 249 

94 864 

95 481 

31 

96 100 

96 721 

97 344 

97 969 

98 596 

99 225 

99 856 

100 489 

101 124 

101 761 

32 

102 400 

103 041 

103 684 

104 329 

104 976 

105 625 

106 276 

106 929 

107 584 

108 241 

33 

108 900 

109 561 

110 224 

110 889 

111 556 

112 225 

112 896 

113 569 

114 244 

114 921 

34 

115 600 

116 281 

116 964 

117 649 

118 336 

119 025 

119716 

120 409 

121 104 

121 801 

35 

122 500 

123 201 

123 904 

124 609 

125 316 

126 025 

126 736 

127 449 

128 164 

128 881 

36 

129 600 

130 321 

131 044 

131 769 

132 496 

133 225 

133 956 

134 689 

135 424 

136 161 

37 

136 900 

137 641 

138 384 

139129 

139 876 

140 625 

141 376 

142 129 

142 884 

143 641 

38 

144 400 

145 161 

145 924 

146 689 

147 456 

148 225 

148 996 

149 769 

150 544 

151 321 

39 

152 100 

152 881 

153 664 

154 449 

155 236 

156 025 

156 816 

157 609 

158 404 

159 201 

40 

160 000 

160 801 

161 604 

162 409 

163 216 

164 025 

164 836 

165 649 

166 464 

167 281 

41 

168100 

168 921 

169 744 

170 569 

171 396 

172 225 

173 056 

173 889 

174 724 

175 561 

42 

176 400 

177 241 

178 084 

178 929 

179 776 

180 625 

181 476 

182 329 

183184 

184 041 

43 

184 900 

185 761 

186 624 

187 489 

188 356 

189 225 

190 096 

190 9691191 844 

192 721 

44 

193 600 

194 481 

195 364 

196 249 

197 136 

198 025 

198 916 

199 809 200 704 

201 601 

45 

202 500 

203 401 

204 304 

205 209 

206 116 

207 025 

207 936 

208 849,209 764 

210 681 

46 

211 600 

212 521 

213 444 

214 369 

215 296 

216 225 

217 156 

218 089 219 024 

219 961 

47 

220 900 

221 841 

222 784 

223 729 

224 676 

225 625 

226 576 

227 529 228 484 

229 441 

48 

230 400 

231 361 

232 324 

233 289 

234 256 

235 225 

236196 

237169 

,23S144 

239 121 

49 

240 100 

241 081 

242 064 

243 049 

244 036 

245 025 

246 016 

247 009 

248 004 

249 001 

50 

250000 

251 001 

252 004 

253 009 

254 016 

255 025 

256 036 

257 049 

258 064 

259 081 

51 

260 100 

261 121 

262144 

263 169 

264 196 

265 225 

266 256 

267 289 

268 324 

269 361 

52 

270 400 

271 441 

272 484 

273 529 

274 576 

275 625 

276 676 

277 729 

278 784 

279 841 

53 

280 900 

281 961 

283 024 

284 089 

285 156 

286 225 

287 296 

288 369 

289 444 

290 521 

54 

291 600 

292 681 

293 764 

294 849 

295 936 

297 025 

298 116 

299 209 

300 304 

301 401 

55 

302 500 

303 601 

304 704 

305 809 

306 916 

30« 025 

309 136 

310 249 

311 364 

312 481 

56 

313 600 

314 721 

315 844 

316 969 

318 096 

319 225 

320 356 

321 489 

322 624 

323 761 

57 

324 900 

326 041 

327 184 

328 329 

329 476 

330 625 

331 776 

332 929 

334 084 

335 241 

58 

336 400 

337 561 

338 724 

339 889 

341 056 

342 225 

343 396 

344 569 

345 744 

346 921 

59 

348 100 

349 281 

350 464 

351 649 

352 836 

354 025 

355 216 

356 409 

357 604 

358 801 


































145 


TABLE IV.-SQUARES OF NUMBERS. -{Concluded., 




* 0 

* * 1 

* 

n g 

* * 3 

* s| 

1=4 

* * 5 

* 

* 6 

* 

* 7 

* * 8 

* 

*11 

60 

360 

000 

361 201 

362 

404 

363 609 

364 

816 

366 

025 

367 

236 

368 449 

369 

664 

370 

881 

61 

372 

100 

373 321 

374 

544 

375 769 

376 

996 

378 

225 

379 

456 

380 

689 

381 

924 

383 

161 

62 

384 

400 

385 641 

386 

884 

388129 

389 

376 

390 

625 

391 

876 

393 

129 

394 

384 

395 

641 

63 

396 

900 

398 161 

399 424 

400 689 

401 

956 

403 

225 

404 

496 

405 

769 

407 

044 

408 

321 

64 

409 

600 

410 881 

412 

164 

413 449 

414 

736 

416 

025 

417 

316 

418 

609 

419 

904 

421 

201 

65 

422 

500 

423 801 

425 

104 

426 409 

427 

716 

429 

025 

430 

336 

431 

649 

432 

964 

434 

281 

66 

435 

600 

436 921 

438 

244 

439 569 

440 

896 

442 

225 

443 

556 

444 

889 

446 

224 

447 

561 

67 

448 

900 

450 241 

451 

584 

452 929 

454 

276 

455 

625 

456 

976 

458 

329 

459 

684 

461 

041 

68 

462 

400 

463 761 

465 

124 

466 489 

467 

856 

469 

225 

470 596 

471 

969 

473 

344 

474 

721 

69 

476 

100 

477 481 

478 

864 

480 249 

481 

636 

483 

025 

484 

416 

485 

809 

487 

204 

488 

601 

70 

490 

000 

491 401 

492 

S04 

494 209 

495 

616 

497 

025 

498 

436 

499 

849 

501 

264 

502 

681 

71 

504 

100 

505 521 

506 

944 

508 369 

509 

796 

511 

225 

512 

656 

514 

089 

515 

524 

516 

961 

72 

518 

400 

519 841 

521 

284 

522 729 

524 

176 

525 

625 

527 

076 

528 

529 

529 

984 

531 

441 

73 

532 

900 

534 361 

535 

824 

537 289 

538 

756 

540 

225 

541 

696 

543 

169 

544 

644 

546 

121 

74 

547 

400 

549 081 

550 

564 

552 049 

553 

536 

555 

025 

556 

516 

55S 

009 

559 

504 

561 

001 

75 

562 

500 

564 001 

565 

504 

567 009 

568 

516 

570 

025 

571 

536 

573 

049 

574 

564 

576 

081 

76 

577 

600 

579 121 

580 

644 

582 169 

583 

696 

585 

225 

586 

756 

588 

289 

589 

824 

591 

361 

77 

592 

900 

594 441 

595 

984 

597 529 

599 076 

600 

625 

602 

176 

603 

729 

605 

284 

606 

841 

78 

608 

400 

609 961 

611 

524 

613 089 

614 

656 

616 

225 

617 

796 

619 

369 

620 

944 

622 

521 

79 

624 

100 

625 681 

627 

264 

628 849 

630 

436 

632 

025 

633 

616 

635 

209 

636 

804 

638 

401 

80 

640 

000 

641 601 

643 

204 

644 809 

646 

416 

648 

025 

649 

636 

651 

249 

652 864 

654 

481 

81 

656 

100 

657 721 

659 

344 

660 969 

662 

596 

664 

225 

665 

856 

667' 

489 

669 

124 

670 

761 

82 

672 

400 

674 041 

675 

684 

677 329 

678 

976 

680 

625 

682 

276 

683 

929 

685 

584 

687 

241 

83 

688 

900 

690 561 

692 

224 

693 889 

695 

556 

697 

225 

698 

896 

700 

569 

702 

244 

703 

921 

84 

705 

600 

707 281 

708 

964 

710 649 

712 

336 

714 

025 

715 

716 

717 

409 

719 

104 

720 

801 

85 

722 

500 

7 24 201 

725 

904 

727 609 

729 

316 

731 

025 

732 

736 

734 

449 

736 

164 

737 881 

86 

739 

600 

741 321 

743 

044 

744 769 

746 

496 

748 

225 

749 

956 

751 

689 

753 

424 

755 

161 

87 

756 

900 

758 641 

760 

384 

762 129 

763 

876 

765 

625 

767 

376 

769 

129 

770 

884 

772 

641 

88 

774 

400 

776 161 

777 

924 

779 689 

781 

456 

783 

225 

784 

996 

786 

769 

788 

544 

790 

321 

89 

792 

100 

793 881 

795 

664 

797 449 

799 

236 

801 

025 

802 

816 

804 

609 

806 

404 

808 

201 

90 

810 

000 

811801 

813 604 

815 409 

817 

216 

819 

025 

820 836 

822 649 

824 

464 

826281 

91 

828 

100 

829 921 

831 

744 

833 569 

835 

396 

837' 

225 

839 

056 

840 

889 

842 

724 

844 

561 

92 

846 

400 

848 241 

850 

084 

851 929 

853 

776 

855 

625 

857 

476 

859 

329 

861 

184 

863041 

93 

864 

900 

866 761 

868 

624 

870489 

872 

356 

874 

225 

876 

096 

877 

969 

879 

844 

881 

721 

94 

883 

600 

885 481 

887 

364- 

889 249 

891 

136 

893 

025 

894 

916 

896 

809 

898 

704 

900 

601 

95 

902 

500 

904 401 

906 

304 

908 209 

910 

116 

912 

025 

913 

936 

915 

849 

917 

764 

919 

681 

96 

921 

600 

923 521 

925 

444 

927 369 

929 

296 

931 

225 

933 

156 

935 

089 

937 

024 

938 

961 

97 

940 

900 

942 841 

944 

784 

946 729 

948 

676 

950 

625 

952 

576 

954 

529 

956 

484 

958 

441 

98 

960 

400 

962 361 

964 

324 

966 289 

968 

256 

970 

225 

972 

196 

974 

169 

976 

144 

978 

121 

99 

980 

100 

982 081 

984 

064 

986 049 

988 

036 

990 

025 

992 

016 

994 

009 

996 

004 

998 

1:01 


) 

























146 


TABLE V.—LOGARITHMS OF NUMBERS. 



0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Diff. 

10 

0000 

0043 

0086 

0128 

0170 

0212 

0253 

0294 

0334 

0374 

42 

11 

414 

453 

492 

531 

569 

607 

645 

682 

719 

755 

38 

12 

792 

828 

864 

899 

934 

969 

1004 

1038 

1072 

1106 

35 

13 

1139 

1173 

1206 

1239 

1271 

1303 

335 | 

367 

399 

430 

32 

14 

461 

492 

523 

553 

584 

614 

644 

673 

703 

73* 

30 

15 

761 

790 

818 

847 

875 

903 , 

931 

959 

987 

2014 

28 

16 

2041 

2068 

2095 

2122 

2148 

2175 ' 

.'01 

2227 

2253 

279 

26 

17 

304 

330 

355 

380 

405 

430 i 

455 

480 

504 

529 

25 

48 

553 

577 

601 

625 

648 

672 

695 

718 1 

742 

765 

24 

19 

788 

810 

833 

856 

878 

900 ] 

923 

945 

967 

989 

22 

20 

3010 

3032 

3054 

3075 

3096 

3118 

3139 

3160 

3181 

3201 

21 

21 

222 

243 

263 

284 

304 

324 

345 

365 

385 

404 

20 

22 

424 

444 

464 

483 

502 

522 

541 

560 

579 

598 

19 

23 

617 

636 

655 

674 

692 

711 

729 

747 

766 

784 


24 

802 

820 

838 

856 

874 

892 

909 

927 

945 

962 

18 

25 

979 

997 

4014 

4031 

4048 

4065 

4082 

4099 

4116 

4133 

17 

26 

4150 

4166 

183 

200 

216 

232 

249 

265 

281 

298 

16 

27 

314 

330 

346 

362 

378 

393 

409 

425 

440 

456 


28 

472 

487 

502 

518 

533 

. 548 

564 

579 

594 

609 

15 

29 

624 

639 

654 

669 

683 

698 

713 

728 

742 

757 


30 

771 

786 

800 

814 

829 

843 

857 

871 

886 

900 

14 

31 

914 

928 

942 

955 

969 

983 

997 

5011 

5024 

5038 


32 

5051 

5065 

5079 

5092 

5105 

5119 

5132 

145 

159 

172 

13 

33 

185 

198 

211 

224 

237 

250 

263 

276 

289 

302 


34 

315 

328 

340 

353 

366 

378 

391 

403 

416 

428 


35 

441 

453 

465 

478 

490 

502 

515 

527 

539 

551 

12 

36 

563 

575 

587 

599 

611 

623 

635 

647 

658 

670 


37 

682 

694 

705 

717 

729 

740 

752 

763 

775 

786 


38 

798 

809 

821 

832 

843 

855 

866 

877 

888 

899 


39 

911 

922 

933 

944 

955 

966 

977 

988 

999 

6010 

11 

40 

6021 

6031 

6042 

6053 

6064 

6075 

6085 

6096 

6107 

117 


41 

128 

138 

149 

159 

170 

180 

191 

201 

212 

222 


42 

232 

243 

253 

263 

274 

284 

294 

304 

314 

325 


43 

335 

345 

355 

365 

375 

385 

395 

405 

415 

425 

10 

44 

435 

444 

454 

464 

474 

484 

493 

503 

513 

522 


45 

532 

542 

551 

561 

571 

580 

590 

599 

609 

618 


46 

628 

637 

646 

656 

665 

675 

684 

693 

702 

712 


47 

721 

730 

739 

749 

758 

767 

776 

785 

794 

803 


48 

812 

821 

830 

839 

848 

857 

866 

875 

884 

893 

9 

49 

902 

911 

920 

928 

937 

946 

955 

964 

972 

981 


50 

990 

998 

7007 

7016 

7024 

7033 

7042 

7050 

7059 

7067 


51 

7076 

7084 

093 

101 

110 

118 

126 

135 

143 

152 


52 

160 

168 

177 

185 

193 

202 

210 

218 

226 

235 


53 

243 

251 

259 

267 

275 

284 

292 

300 

308 

316 

8 

54 

324 

332 

340 

348 

356 

364 

372 

380 

388 

396 



0 

1 

2 

3 

4 

5 

6 

7 

8 

1 9 








































147 




TABLE V.—LOGARITHMS OF NUMBERS.-(ConchtdeeL) 




0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Diff 


55 

7404 

7412 

7419 

7427 

7435 

7443 

7451 

7459 

7466 

7474 

8 


56 

482 

490 

497 

505 

513 

520 

528 

536 

543 

551 



57 

559 

566 

574 

582 

589 

597 

604 

612 

619 

627 



58 

634 

642 

649 

657 

664 

672 

679 

686 

694 

701 



59 

709 

716 

723 

731 

738 

745 

752 

760 

767 

774 

- 


60 

782 

789 

796 

803 

810 

818 

825 

832 

839 

846 



61 

853 

860 

868 

875 

882 

889 

896 

903 

910 

917 

7 


62 

924 

931 

938 

945 

952 

959 

966 

973 

980 

987 



63 

993 

8000 

8007 

8014 

8021 

8028 

8035 

8041 

8048 

8055 



64 

8062 

069 

075 

082 

089 

096 

102 

109 

116 

122 



65 

129 

136 

142 

149 

156 

162 

169 

176 

182 

189 



66 

195 

202 

209 

• 215 

222 

228 

235 

241 

248 

254 



67 

261 

267 

274 

280 

287 

293 

299 

306 

312 

319 



68 

325 

331 

338 

344 

351 

357 

363 

370 

376 

382 



69 

388 

395 

401 

407 

414 

420 

426 

432 

439 

445 



70 

451 

457 

463 

470 

476 

482 

488 

494 

500 

506 



71 

513 

519 

525 

531 

537 

543 

549 

555 

561 

567 

6 


72 

573 

579 

585 

591 

597 

603 

609 

615 

621 

627 



73 

633 

639 

645 

651 

657 

663 

669 

675 

681 

686 



74 

692 

698 

704 

710 

716 

722 

727 

733 

739 

745 



75 

751 

756 

762 

768 

774 

779 

785 

791 

797 

802 



76 

808 

814 

820 

825 

831 

837 

842 

848 

854 

859 



77 

865 

871 

876 

882 

887 

893 

899 

904 

910 

915 



78 

921 

927 

932 

938 

943 

949 

954 

960 

965 

971 



79 

976 

982 

987 

993 

998 

9004 

9009 

9015 

9020 

9025 



80 

9031 

9036 

9042 

9047 

9053 

058 

063 

069 

074 

079 



• 81 . 

085 

090 

096 

101 

106 

112 

117 

122 

128 

133 



82 

138 

143 

149 

154 

159 

165 

170 

175 

180 

186 



83 

191 

196 

201 

206 

212 

217 

222 

227 

232 

238 



84 

243 

248 

253 

258 

263 

269 

274 

279 

284 

289 



85 

294 

299 

304 

309 

315 

320 

325 

330 

335 

340 



86 

345 

350 

355 

360 

365 

370 

375 

380 

385 

390 

5 


87 

395 

400 

405 

410 

415 

420 

425 

430 

435 

440 



88 

445 

450 

455 

460 

465 

469 

474 

479 

484 

489 



89 

494 

499 

504 

509 

> 

513 

518 

523 

528 

533 

538 



90 

542 

547 

552 

557 

562 

566 

571 

576 

581 

586 



91 

590 

595 

600 

■605 

609 

614 

619 

624 

628 

633 



92 

638 

643 

647 

652 

657 

661 

666 

671 

675 

680 



93 

685 

689 

694 

699 

703 

708 

713 

717 

722 

727 



94 

731 

736 

741 

745 

750 

754 

759 

763 

768 

773 



95 

777 

782 

786 

791 

795 

800 

805 

809 

814 

818 



96 

823 

827 

832 

836 

841 

845 

850 

854 

859 

863 



97 

868 

872 

877 

881 

886 

890 

894 

899 

903 

908 



. 98 

912 

917 

921 

926 

930 

934 

939 

943 

948 

952 



99 

956 

961 

965 

969 

974 

978 

983 

987 

991 

996 

4 



0 

1 

2 

3 • 

4 

5 

6 

7 

8 

9 





















































148 


TABLE VI.—LOGARITHMIC TRIGONOMETRIC FUNCTIONS. 



0 ° 

1 ° 

2 ° 


' 

Sin 

Tan 

Cot 

Sin 

Tan 

Cot 

Sin 

Tan 

Cot 

/ 

0 

— 00 

-f* co 

8.2419 

8.2419 

11.7581 

8.5428 

8.5431 

11.4569 

60 

1 

6.4637 

13.5363 

2490 

2491 

7509 

5464 

5467 

4533 

59 

o 

7648 

2352 

2561 

2562 

7438 

5500 

5503 

4497 

58 

3 

9408 

0592 

2630 

2631 

7369 

5535 

5538 

4462 

57 

4 

7.0658 

12.9342 

2699 

2700 

7300 

5571 

5573 

4427 

56 

5 

1627 

8373 

2766 

2767 

7233 

5605 

5608 

4392 

55 

(5 

2419 

7581 

2832 

2833 

7167 

5640 

5643 

4357 

54 

7 

3088 

6912 

2898 

2899 

7101 

5674 

5677 

4323 

53 

8 

3668 

6332 

2962 

2963 

7037 

5708 

. 5711 

4289 

52 

9 

4180 

5820 

3025 

3026 

6974 

5742 

5745 

4255 

51 

10 

4637 

5363 

3088 

3089 

6911 

5776 

5779 

4221 

50 

11 

7.5051 

12.4949 

8.3150 

8.3150 

11.6850 

8.5809 

8.5812 

11.4188 

49 

12 

5429 

4571 

3210 

3211 

6789 

5842 

5845 

4155 

48 

13 

5777 

4223 

3270 

3271 

6729 

5875 

5878 

4122 

47 

14 

6099 

3901 

3329 

3330 

667Q 

5907 

5911 

4089 

46 

15 

6398 

3602 

3388 

3389 

6611 

5939 

5943 

4057 

45 

16 

6678 

3322 

3445 

3446 

6554 

5972 

5975 

4025 

44 

17 

6942 

3058 

3502 

3503 

6497 

6003 

6007 

3993 

43 

18 

7190 

2810 

3558 

3559 

6441 

6035 

6038 

3962 

42 

19 

7425 

2575 

3613 

3614 

6386 

6066 

6070 

3930 

41 

20 

7648 

2352 

3668 

3669 

6331 

6097 

6101 

3899 

40 

21 

7.7859 

12.2140 

8.3722 

8.3723 

11.6277 

8.6128 

8.6132 

11.3868 

39 

22 

8061 

1938! 

3775 

3776 

6224 

6159 

6163 

3837 

38 

23 

8255 

1745 

3828 

3829 

6171 

6189 

6193 

3807 

37 

24 

8439 

1561! 

3880 

3881 

6119 

6220 

6223 

3777 

36 

25 

8617 

1383 

3931 

3932 

6068 

6250 

6254 

3746 

35 

26 

8787 

1213 

3982 

3983 

6017 

6279 

6283 

3717 

34 

27 

8951 

1049 

4032 

4033 

5967 

6309 

6313 

■3687 

33 

28 

9109 

0891 

4082 

4083 

5917 

6339 

6343 

3657 

32 

29 

9261 

0739 

4131 

4132 

5868 

6368 

6372 

3628 

31 

30 

9408 

0591 

4179 

4181 

5819 

6397 

6401 

3599 

30 

31 

7.9551 

12.0449 

8.4227 

8.4229 

11.5771 

8.6426 

8.6430 

11.3570 

29 

32 

9689 

0311 

4275 

4276 

5724 

6454 

6459 

3541 

28 

33 

9822 

0177 

4322 

4323 

56771 

6483 

6487 

3513 

27 

34 

9952 

0048 

4368 

4370 

5630 

6511 

6515 

3485 

26 

35 

8.0078 

11.9922 

4414 

4416 

5584 

6539 

6544 

3456 

25 

36 

0200 

9800 

4459 

4461 

5539 

6567 

6571 

3429 

24 

37 

0319 

9681 

4504 

4506 

5494 

6595 

6599 

3401 

23 

38 

0435 

9565 

4549 

4551 

5449 

6622 

6627 

3373 

22 

39 

0548 

9452 

4593 

4595 

5405 

6650 

6654 

3346 

21 

40 

0658 

9342 

4637 

4638 

5362 

6677 

6682 

3318 

20 

/ 

Cos 

Cot 

Tan 

Cos 

Cot 

Tan 

Cos 

Cot 

Tan 

/ 


89° 

88 ° 

8; ° 




































149 


f ABLE VI.—LOGARITHMIC TRIGONOMETRIC FUNCTIONS.-(Con^m<e<Z.) 



0° 

1° 

2° 


/ 

Sin 

Tan 

Cot 

Sin 

Tan 

Cot 

Sin 

Tan 

Cot 

f 

41 

8.0765 

11.9235 

8.4680 

8.4682 

11.5318 

8.6704 

8.6709 

11.3291 

19 

42 

0870 

9130 

4723 

4725 

5275 

6731 

6736 

3264 

18 

43 

0972 

9028 

4765 

4767 

5233 

6758 

6762 

3238 

17 

44 

1072 

8928 

4807 

4809 

5191 

6784 

6789 

3211 

16 

45 

1169 

8830 

4848 

4850 

5150 

6810 

6815 

3185 

15 

46 

1265 

8735 

4890 

4892 

5108 

6837 

6842 

3158 

14 

47 

1358 

8641 

4930 

4932 

5068 

6863 

6868 

3132 

13 

48 

1450 

8550 

4971 

4973 

5027 

6889 

6894 

3106 

12 

49 

1539 

8460 

5011 

5013 

4987 

6914 

6920 

3080 

11 

50 

1627 

8313 

5050 

5053 

4947 

6940 

6945 

3055 

10 

51 

8.1713 

11.8287 

8.5090 

8.5092 

11.4908 

8.6965 

8.6971 

11.3029 

9 

52 

1797 

8202 

5129 

5131 

4869 

6991 

6996 

3004 

8 

53 

1880 

8120 

5167 

5170 

i 4830 

7016 

7021 

2979 

7 

54 

1961 

8038 

5206 

5208 

4792 

7041 

7046 

2954 

6 

55 

2041 

7959 

5243 

5246 

4754 

7066 

7071 

2929 

5 

56 

2119 

7880 

5281 

5283 

4717 

7090 

7096 

2904 

4 

57 

2196 

7804 

5318 

5321 

4679 

7115 

7121 

2879 

3 

58 

2271 

7728 

5355 

5358 

4642 

7140 

7145 

2855 

2 

59 

2346 

7654 

5392 

5394 

4606 

7164 

7170 

2830 

1 

60 

8.2419 

11.7581 

8.5428 

8.5431 

11.4569 

8.7188 

8.7194 

11.2806 

0 

/ 

Cos 

Cot 

Tan 

Cos 

Cot 

Tan 

Cos 

Cot 

Tail 

/ 


89° 

88° 

87° 



Cosines. 0° to 3°. 


/ 

0° 

1° 

2° 

/ 

0 

10.0000 

9.9999 

9.9997 

60 

10 

0000 

9999 

9997 

50 

20 

0000 

9999 

9996 

40 

30 

0000 

9999 

9996 

30 

40 

0000 

9998 

9995 

20 

50 

0000 

9998 

9995 

10 

60 

9.9999 

9.9997 

9.9994 

0 

/ 

89° 

88° 

oo 

•a 

o 

/ 


I 


Sines. 87° to 90°. 



















































150 


TABLE VI—LOGARITHMIC TRIGONOMETRIC FUNCTIONS. -(Continued. 


o / 

Sin 

Diff. 

1 ' 

Cos 

Tan 

Diff. 

1 ' 

Cot 

O / 

3° 

8.7188 

23.5 

22.3 
21.2 
20.2 

19.3 

18.5 

9.9994 

8-7194 

23 5 

22.3 
21.2 

20.3 

19.4 

18.5 

11.2806 

87° 

10 

7423 

9993 

7429 

2571 

50 

20 

7645 

9993 

7652 

2348 

40 

30 

7857 

9992 

7865 

2135 

30 

40 

8059 

9991 

8067 

1933 

20 

50 

8251 

9990 

8261 

1739 

10 

4° 

8.8436 

17.7 
17.0 
16.3 

15.8 
15.2 
14.7 

9.9989 

8.8446 

17.8 
17.1 
16.5 

15.8 
15.3 

14.8 

11.1554 

86° 

10 

8613 

9989 

8624 

1376 

50 

20 

8783 

9988 

8795 

1205 

40 

30 

8946 

9987 

8960 

1040 

30 

40 

9104 

9986 

9118 

0882 

20 

50 

9256 

9985 

9272 

0728 

10 

5° 

8.9403 

14.2 
13.7 

13.4 
12.9 

12.5 

12.2 

9.9983 

8.9420 

14.3 
13.8 

13.4 
13.0 
12.7 
12.3 

11.0580 

85° 

10 

9545 

9982 

9563 

0437 

50 

20 

9682 

9981 

9701 

0299 

40 

30 

9816 

9980 

9836 

0164 

30 

40 

9945 

9979 

9966 

0034 

20 

50 

9.0070 

9977 

9.0093 

10.9907 

10 

6° 

9.0192 

11.9 
11.5 
11.2 

10.9 
10.7 
10.4 

9.9976 

9.0216 

12.0 

11.7 

11.4 
11.1 

10.8 

10.5 

10.9784 

84° 

10 

0311 

9975 

0336 

9664 

50 

20 

0426 

9973 

0453 

9547 

40 

30 

0539 

9972 

0567 

9433 

30 

40 

0648 

9971 

0678 

9322 

20 

50 

0755 

9969 

0786 

9214 

10 

7° 

9.0859 

10.2 

9.9 

9.7 

9.5 

9.3 

9.1 

9.9968 

9.0891 

10.4 

10.1 

9.8 

9.7 

9.4 

9.3 

10.9109 

83° 

10 

0961 

9966 

0995 

9005 

50 

20 

1060 

9964 

1096 

8904 

40 

30 

1157 

9963 

1194 

8806 

30 

40 

1252 

9961 

1291 

8709 

20 

50 

1345 

9959 

1385 

8615 

10 

8° 

9.1436 

8.9 
8.7 
8.5 1 
8.4 
8.2 
8.0 

9.9958 

9.1478 

9.1 
8.9 
8.7 
8.6 
8.4 

8.2 

10.8522 

82° 

10 

1525 

9956 

1569 • 

8431 

50 

20 

1612 

9954 

1658 

8342 

40 

30 

1697 

9952 

1745 

8255 

30 

40 

1781 

9950 

1831 

8169 

20 

50 

1863 

9948 

1915 

8085 

10 

9° 

9.1943 

7.9 

7.8 

7.6 

7.5 

7.3 

7.2 

9.9946 

9.1997 

8.1 

8.0 

7.8 

7.7 

7.6 

7.4 

10.8003 

81° 

10 

2022 

9944 

2078 

7922 

50 

20 

2100 

9942 

2158 

7842 

40 

30 

2176 

9940 

2236 

7764 

30 

40 

2251 

9938 

2313 

7687 

20 

50 

2324 

9936 

2389 

7611 

10 

10* 

9.2397 


9.9934 

9.2463 


10.7537 

80° 

o / 

Cos 

Diff. 
1' ! 

Sin 

Cot 

Diff. 

V 

Tan 

O J 






































151 


TABLE VT.—LOGARITHMIC TRIGONOMETRIC FUNCTIONS.— (Continue^ 
l-;--- 


• / 

Sin 

Diff. 

V 

Cos 

Tan 

Diff. 

V 

Cot 

O 

10° 

9.2397 

7.1 

7.0 

6.9 

6.8 

6.7 

6.6 

9.9934 

9.2463 

7.3 

7.3 

7.1 

7.0 

6.9 

6.8 

10.7537 

80° 

10 

2468 

9931 

2536 

7464 

50 

20 

2538 

9929 

2609 

7391 

40 

30 

2606 

9927 

2680 

7320 

30 

40 

2674 

9924 

2750 

7250 

20 

50 

2740 

9922 

2819 

7181 

10 

O 

H 

9.2806 

6.4 

6.4 

6.3 

6.2 

6.1 

6.0 

9.9919 

9.2887 

6.7 

6.6 

6.5 

6.4 

6 3 
6.2 

10.7113 

79° 

10 

2870 

9917 

2953 

7047 

50 

20 

2934 

9914 

3020 

6980 

40 

30 

2997 

9912 

3085 

6915 

30 

40 

3058 

9909 

3149 

6851 

20 

50 

3119 

9907 

3212 

6788 

10 

13° 

9.3179 

5.9 

5.8 

5.7 

5.7 

5.6 

5.5 

9.9904 

9.3275 

6.1 

6.1 

6.0 

5.9 

5.9 

5.8 

10.6725 

78° 

10 

3238 

9901 

3336 

6664 

50 

20 

3296 

9899 

3397 

6603 

40 

30 

3353 

9896 

3458 

6542 

30 

40 

3410 

9893 

3517 

6483 

20 

50 

3466 

9890 

3576 

6424 

10 

13° 

9.3521 

*5.4 

5 4 
5.3 
5.2 
5.2 
5.1 

9.9887 

9.3634 

K 

10.6366 

77° 

10 

3575 

9884 

3691 

O. i 

5.7 

5.6 

5.5 

5.5 

5.4 

6309 

50 

. 20 

3629 

9881 

3748 

6252 

40 

30 

3682 

9878 

3804 

6196 

30 

40 

3734 

9875 

3859 

6141 

20 

50 

3786 

9872 

3914 

6086 

10 

14° 

9.3837 

5.0 

5.0 

4.9 

4.9 

4.8 

4.7 

9.9869 

9.3968 

5.3 

5.3 

5.2 

5.2 

5.1 

5.1 

10.6032 

76° 

10 

3887 

9866 

4021 

5979 

50 

20 

3937 

9863 

4074 

5926 

40 

30 

3986 

9859 

4127 

5873 

30 

40 

4035 

9856 

4178 

5822 

20 

50 

4083 

9853 

4230 

5770 

10 

15° 

9.4130 

4.7 

4.6 

4.6 

4.5 

4.5 

4.4 

9.9849 

9.4281 

5.0 

5.0 

4.9 

4.9 

4.8 

4.8 

10.5719 

75° 

10 

4177 

9846 

4331 

5669 

50 

20 

4223 

9843 

4381 

5619 

40 

30 

4269 

9839 

4430 

5570 

30 

40 

4314 

9836 

4479 

5521 

20 

50 

4359 

9832 

4527 

5473 

10 

16° 

9.4403 

4.4 

4.4 

4.3 

4.2 

4.2 

4.1 

9.9828 

9.4575 

4.7 

4.7 

4.7 

4.6 

4.6 

4.5 

10.5425 

74° 

10 

4447 

9825 

• 4622 

5378 

50 

20 

4491 

9821 

4669 

5331 

40 

30 

4533 

9817 

4716 

•.5284 

30 

40 

4576 

9814 

4762 

5238 

20 

50 

4618 

9810 

4808 

5192 

10 

H 

1 ^ 

0 

9.4659 


9.9806 

9.4853 


10.5147 

73° 

o / 

Cos 

Diff. 

V 

Sin 

Cot 

Diff. 

V 

Tan 

O / 






























152 


TABLE VI.—LOGARITHMIC TRIGONOMETRIC FUNCTIONS.— (Continued.) 


o / 

Sin 

Diff. 

V 

Cos 

Tan 

Diff. 

V 

Cot 

O / 

17° 

9.4659 

4.1 

4.1 

4.0 

4.0 

4.0 

3.9 

9.9806 

9.4853 

4.5 

4.5 

4.4 

4.4 

4.4 

4.3 

10.5147 

73° 

10 

4700 

9802 

4898 

5102 

50 

20 

4741 

9798 

4943 

5057 

40 

30 

4781 

9794 

4987 

5013 

30 

40 

4821 

9790 

5031 

4969 

20 

50 

4861 

9786 

5075 

4925 

10 

18° 

9.4900 

3.9 

3.8 

3.8 

3.8 

3.7 

3.7 

9.9782 

9.5118 

4.3 

4.2 

4.2 

4.2 

4.2 

4.1 

10.4882 

72° 

10 

4939 

9778 

5161 

4839 

50 

20 

4977 

9774 

5203 

4797 

40 

30 

5015 

9770 

5245 

4755 

30 

40 

5052 

9765 

5287 

4713 

20 

50 

5090 

9761 

5329 

4671 

10 

19° 

9.5126 

3.7 

3.6 

3.6 

3.6 

3.5 

3.5 

9.9757 

9.5370 

4.1 

4.0 

4.0 

4.0 

4.0 

4.0 

10.4630 

71° 

10 

5163 

9752 

5411 

4589 

50 

20 

5199 

9748 

5451 

4549 

40 

30 

5235 

9743 

5491 

4509 

30 

40 

5270 

9739 

5531 

4469 

20 

50 

5306 

9734 

5571 

4429 

10 

20° 

9.5341 

3.5 

3.4 

3.4 

3.4 

3.3 

3.3 

9.9730 

9.5611 ' 

•S.9 

3.9 

3.9 

3.8 

3.8 

3.8 

10.4389 

70° 

10 

5375 

9725 

5650 

4350 

50 

20 

5409 

9721 

5689 

4311 

40 , 

30 • 

5443 

9716 

5727 

4273 

30 

40 

5477 

9711 

5766 

4234 

20 

50 

5510 

9706 

5804 

4196 

10 

21° 

9.5543 

3.3 

3.3 

3.2 

3.2 

3.2 

3.1 

9.9702 

9.5842 

3.8 

3.8 

3.7 

3.7 

3.7 

3.6 

10.4158 

69° 

10 

5576 

9697 

5879 

4121 

50 

20 

5609 

9692 

5917 

4083 

40 

30 

5641 

9687 

5954 

4046 

30 

40 

5673 

9682 

5991 

4009 

20 

50 

5704 

9677 

6028 

3972 

10 

22° 

9.5736 

3.1 

3.1 

3.1 

3.0 

3.0 

3.0 

9.9672 

9.6064 

3.6 

3.6 

3.6 

3.6 

3.5 

3.5 

10.3936 

68° 

10 

5767 

9667 

6100 

3900 

50 

20 

5798 

9661 

6136 

3864 

40 

30 

5828 

9656 

6172 

3828 

30 

40 

5859 

9651 

6208 

3792 

20 

50 

5889 

9646 

6243 

3757 

10 

23° 

9.5919 

3.0 

2.9 

2.9 

2.9 

2.9 

2.8 

9.9640 

9.6279 

3.5 

3.5 

3.5 

3.4 

3.4 

3.4 

10.3721 

6,7° 

10 

5948 

9635 

6314 

3686 

50 

20 

5978 

9629 

6348 

3652 

40 

30 

6007 

9624 

6383 

3617 

30 

40 

6036 

9618 

6417 

3583 

20 

50 

6065 

9613 

6452 

3548 

10 

24° 

9.6093 


9.9607 

9.6486 


10.3514 

66° 

O / 

Cos 

Diff. 

1 ' 

Sin 

Cot 

Diff. 

V 

Tan 

o i 
























153 


TABLE VI.—LOGARITHMIC TRIGONOMETRIC FUNCTIONS.— (Continued.\ 


o / 

Sin 

Dili. 

V 

Cos 

24° 

10 

iiO 

30 

40 

50 

25° 

10 

20 

30 

40 

50 

26° 

10 

20 

30 

40 

50 

27° 

10 

20 

30 

40 

50 

28° 

10 

20 

30 

40 

50 

29° 

10 

20 

30 

40 

50 

30° 

10 

20 

30 

40 

50 

31° 

9.6093 

6121 

6149 

6117 

6205 

6232 

9.6259 

6286 

6313 

6340 

6366 

6392 

9.6418 

6444 

6470 

6495 

6521 

6546 

9.6570 

6595 

6620 

6644 

6668 

6692 

9.6716 

6740 

6763 

6787 

6810 

6833 

9.6856 

6878 

6901 

6923 

6946 

6968 

9.6990 

7012 

7033 

7055 

7076 

7097 

9.7118 

2.8 

2.8 

2.8 

2.8 

2.7 

2.7 

2.7 

2.7 

2.7 

2.6 

2.6 

2.6 

2.6 

2.6 

2.5 

2.5 

2.5 

2.5 

2.5 

2.5 

2.4 

2.4 

2.4 

2.4 

2.4 

2.4 

2.3 

2.3 

2.3 

2.3 

2.3 

2.3 

2.2 

2.2 

2.2 

2.2 

2.2 

2.2 

2.1 

2.1 

2.1 

2.1 

9.9607 

9602 

9596 

9590 

9584 

9579 

9.9573 

9567 

9561 

9555 

9549 

9543 

9.9537 

9530 

9524 

9518 

9512 

9505 

9.9499 

9492 

9486 

9479 

9473 

9466 

9.9459 

9453 

9446 

9439 

9432 

9425 

9.9418 

9411 

9404 

9397 

9390 

9383 

9.9375 

9368 

9361 

9353 

9346 

9338 

9.9331 

O / 

Cos 

Diff. 

V 

Sin 


Tan 

Diff. 

V 

Cot 

O / 

9.6486 

3.4 

3.4 

3.4 

3.3 

3.3 

3 3 

10.3514 

66 ° 

6520 

3480 

50 

6553 

3447 

40 

6587 

3413 

30 

6620 

3380 

20 

66 'G 

3346 

10 

9.6687 

3.3 

3.3 

3.3 

3.2 

3.2 

3 2 

10.3313 

65° 

6720 

3280 

50 

6752 

3248 

40 

6785 

3215 

30 

6817 

3183 

20 

6850 

3150 

10 

9.6882 

3.2 

3.2 

3 2 
3.2 
3.1 
3.1 

10.3118 

64° 

6914 

3086 

50 

6946 

3054 

40 

6977 

3023 

30 

7009 

2991 

20 

7040 

2960 

10 

9.7072 

3.1 

O 1 

10.2928. 

63° 

7103 

2897 

50 

7134 

3.1 

3.1 

3 1 
3.1 

2866 

40 

7165 

2835 

30 

7196 

2804 

20 

7226 

2774 

10 

9.7257 

3.0 

3.0 

3.0 

3.0 

3.0 

3.0 

10.2743 

62° 

7287 

2713 

50 

7317 

2683 

40 

7348 

2652 

30 

7378 

2622 

20 

7408 

2592 

10 

9.7438 

3.0 

3.0 

3.0 

2.9 

2.9 

2.9 

10.2562 

61° 

7467 

2533 

50 

7497 

2503 

40 

7526 

2474 

30 

7556 

2444 

20 

7585 

2415 

10 

9.7614 

2.9 

2.9 

2.9 

2.9 

2.9 

2.9 

10.2386 

60° 

7644 

2356 

50 

7673 

2327 

40 

7701 

2299 

30 

7730 

7759 

2270 

2241 

20 

10 

9.7788 

1 

10.2212 

59° 

Cot 

Diff. 

V 

Tan 

O / 


f 

























154 


TABLE VI.— LOGARITHMIC TRIGONOMETRIC FUNCTIONS.—( Continued.} 


• / 

Sin 

Diff. 

V 

Cos 

Diff. 

V 

Tan 

Diff. 

V 

Cot 

i 

1 

o / 

31° 

10 

20 

30 

40 

50 

9.7118 

7139 

7160 

7181 

7201 

7222 

2.1 

2.1 

2.1 

2.1 

2.0 

2.0 

9.9331 

9323 

9315 

9308 

9300 

9292 

0.8 

0.8 

0.8 

0.8 

0.8 

0.8 

9.7788 

7816 

7845 

7873 

7902 

7930 

2.9 

2.8 

2.8 

2.8 

2.8 

2.8 

10.2212 

2181 

2155 

212',' 

2098' 

207C 

59° 

50 

40 

30 

20 

10 

33° 

10 

20 

30 

40 

50 

9.7242 

7262 

7282 

7302 

7322 

7342 

2.0 

2.0 

2.0 

2.0 

2.0 

2.0 

9.9284 

9276 

9268 

9260 

9252 

9244 

0.8 

0.8 

0.8 

0.8 

O.S 

0.8 

9.7958 

7986 

8014 

8042 

8070 

8097 

2.8 

2.8 

2.8 

2.8 

2.8 

2.8 

10.2042 

2014 

1986 

1958 

1930 

1903 

58° 

50 

40 

30 

20 

10 

33° 

10 

20 

30 

40 

50 

9.7361 

7380 

7400 

7419 

7438 

7457 

1.9 

1.9 

1.9 

1.9 

1.9 

1.9 

9.9236 

9228 

9219 

9211 

9203 

9194 

0.8 

0.8 

0.8 

0.8 

0.8 

0.8 

9.8125 

8153 

8180 

8208 

8235 

8263 

2.8 

2.8 

2.7 

2.7 

2.7 

2.7 

10.1875 

1847 

1820 

1792 

1765 

1737 

57° 

50 

40 

30 

20 

10 

34° 

10 

20 

30 

40 

50 

9.7476 

7494 

7513 

7531 

7550 

7568 

1.9 

1.9 

1.8 

1.8 

1.8 

1.8 

9.9186 

9177 

9169 

9160 

9151 

9142 

0.9 

0.9 

0.9 

0.9 

0.9 

0.9 

9.8290 

8317 

8344 

8371 

8398 

8425 

2.7 

2.7 

2.7 

2.7 

2.7 

2.7 

10.1710 

1683 

1656 

1629 

1602 

1575 

56° 

50 

40 

30 

20 

10 

35° 

10 

20 

30 

40 

50 

9.7586 

7604 

7622 

7640 

7657 

7675 

1.8 

1.8 

1.8 

1.8 

1,8 

1.7 

9.9134 

9125 

9116 

9107 

9098 

9089 

0.9 

0.9 

0.9 

0.9 

0.9 

0.9 

9.8452 

8479 

8506 

8533 

8559 

8586 

2.7 

2.7 

2.7 

2.7 

2.7 

2.7 

10.1548 

1521 

1494 

1467 

1441 

1414 

55° 

50 

40 

30 

20 

10 

36° 

10 

20 

30 

40 

50 

9.7692 

7710 

7727 

7744 

7761 

7778 

1.7 

1.7 

1.7 

1.7 

1.7 

1.7 

9.9080 

9070 

9061 

9052 

9042 

9033 

0.9 

0.9 

0.9 

0.9 

0.9 

0.9 

9.8613 

8639 

8666 

8692 

8718 

8745 

2.7 

2.6 

2.6 

2.6 

2.6 

2.6 

10.1387 

1361 

1334 

1308 

1282 

1255 

54° 

50 

40 

30 

20 

10 

37° 

10 

20 

30 

40 

50 

9.7795 

7811 

7828 

7844 

7861 

7877 

1.7 

1.7 

1.7 

1.6 

1.6 

1.6 

9.9023 

9014 

9004 

8995 

8985 

8975 

1.0 

1.0 

1.0 

1.0 

1.0 

1.0 

9.8771 

8797 

8824 

8850 

8876 

8902 

2.6 

2.6 

2.6 

2.6 

2.6 

2.6 

10.1229 

1203 

1176 

1150 

1124 

1098 

53° 

50 

40 

30 

20 

10 

38* 

9.7893 


9.8965 


9.8928 


10.1072 

52° 

o / 

Cos 

Diff. 

V 

Sin 

Diff. 

1' 

Cot 

Diff. 

V 

Tan 

O f 











































155 


JAB LB VI— LOGARITHMIC TRIGONOMETRIC FUNCTIONS.- -{Concluded.) 


o / 

Sin 

Diff. 

V 

Cos 

Diff. 

1' 

Tan 

Diff. 

V 

Cot 

O f 

38° 

10 

20 

30 

40 

50 

39° 

10 

20 

30 

40 

50 

40° 

10 

20 

30 

40 

50 

41° 

10 

20 

30 

40 

50 

42° 

10 

20 

30 

40 

50 

43° 

10 

20 

30 

40 

50 

44° 

10 

20 

30 

40 

50 

45° 

9.7893 

7910 

7926 

7941 

7957 

7973 

9.7989 

8004 

8020 

8035 

8050 

8066 

9.8081 

8096 

8111 

8125 

8140 

8155 

9.8169 

8184 

8198 

8213 

8227 

8241 

9.8255 

8269 

8283 

8297 

8311 

8324 

9.8338 

8351 

8365 

8378 

8391 

8405 

9.8418 

8431 

8444 

8457 

8469 

8482 

9.8495 

1.6 

1.6 

1.6 

1.6 

1.6 

1.6 

1.6 

1.5 

1.5 

1.5 

1.5 

1.5 

1.5 

1.5 

1.5 

1.5 

1.5 

1.5 

1.4 

1.4 

1.4 

1.4 

1.4 

1.4 

1.4 

1.4 

1.4 

1.4 

1.4 

1.4 

1.4 

1.3 

1.3 

1.3 

1.3 

1.3 

1.3 
1.3 
1.3 
1.3 
1.3 
. 1.3 

9.8965 
8955 
8945 
8935 
8925 
8915 

9.8905 

8895 

8884 

8874 

8864 

8853 

9.8843 

8832 

8821 

8810 

8800 

8789 

9.8778 

8767 

8756 

•8745 

8733 

8722 

9.8711 

8699 

8688 

8676 

8665 

8653 

9.8641 

8629 

8618 

8606 

8594 

8582 

9.8569 
8557 
8545 
8532 
8520 
8507 ■ 

9.S495 

1.0 

1.0 

1.0 

1.0 

1.0 

1.0 

1.0 

1.0 

1.0 

1.0 

1.1 

1.1 

1.1 

1.1 

1.1 

1.1 

1.1 

1.1 

1.1 

1.1 

1.1 

1.1 

1.1 

1.1 

1.1 

1.1 

1.2 

1.2 

1.2 

1.2 

1.2 

1.2 

1.2 

1.2 

1.2 

1.2 

1.2 

1.2 

1.2 

1.2 

1.3 

1.3 

9.8928 

8954 

8980 

9006 

9032 

9058 

9.9084 

9110 

9135 

9161 

9187 

9212 

9.9238 

9264 

9289 

9315 

9341 

9366 

9.9392 

9417 

9443 

9468 

9494 

9519 

9.9544 

9570 

9595 

9621 

9646 

9671 

9.9697 

9722 

9747 

9772 

9798 

9823 

9.9848 

9874 

9899 

9924 

9949 

9975 

10.0000 

2.6 

2.6 

2.6 

2.6 

2.6 

2.6 

2.6 

2.6 

2.6 

2.6 

2.6 

2.6 

2.6 

2.6 

2.6 

2.6 

2.6 

2.6 

2.6 

2.5 

2.5 

2.5 

2.5 

2.5 

2.5 

2.5 

2.5 

2.5 

2.5 

2.5 

2.5 

2.5 

2.5 

2.5 

2.5 

2.5 

2.5 

2.5 

2.5 

2.5 

2.5 

2.5 

10.1072 

1046 

1020 

• 0994 
0968 
0942 

10.0916 

0890 

0865 

0839 

0813 

0788 

10.0762 

0736 

0711 

0685 

0659 

0634 

10.0608 

0583 

0557 

0532 

0506 

0481 

10.0456 

0430 

0405 

0379 

0354 

0329 

10.0303 

0278 

0253 

0228 

0202 

0177 

10.0152 

0126 

0101 

0076 

0051 

0025 

10.0000 

52° 

50 

40 

30 

20 

10 

51° 

50 

40 

30 

20 

10 

50° 

50 

40 

30 

20 

10 

49° 

50 

40 

30 

20 

10 

48° 

50 

40 

30 

20 

10 

47° 

50 

40 

30 

20 

10 

4G° 

50 

40 

30 

20 

10 

45° 

o / 

Cos 

Diff. 

V 

i 

Sin 

Diff. 

V 

Cot 

Diff. 

V 

Tan 

O / 























































156 


TABLE VII.—NATURAL SINES AND COSINES. 

O 0 —7° 30' 7° 30 —15° 


o / 

Sin. 

Cos. 


0 0 

0.0000 

1.0000 

0 90 

10 

0.0029 

1.0000 

50 

20 

0.0058 

1.0000 

40 

30 

0.0087 

1.0000 

30 

40 

0.0116 

0.9999 

20 

50 

0.0145 

0.9999 

10 

1 0 

0.0175 

0.9998 

0 89 

10 

0.0204 

0.9998 

50 

20 

0.0233 

0.9997 

40 

30 

0.0262 

0.9997 

30 

40 

0.0291 

0.9996 

20 

50 

0.0320 

0.9995 

10 

2 0 

0.0349 

0.9994 

0 88 

10 

0.0378 

0.9993 

50 

20 

0.0407 

0.9992 

40 

30 

0.0436 

0.9990 

30 

40 

0.0465 

0.9989 

20 

50 

0.0494 

0 9988 

10 

3 0 

0.0523 

0.9986 

O 87 

10 

0.0552 

0.9985 

50 

20 

0.0581 

0.9983 

40 

30 

0.0610 

0.9981 

30 

40 

0 0640 

0.9980 

20 

50 

0.0669 

0.9978 

10 

4 0 

0.0098 

0.9976 

O 86 

10 

0.072; 

0.9974 

50 

20 

0.0756 

0.9971 

40 

30 

0.0785 

0 9969 

30 

40 

0.0814 

0.9967 

20 

50 

0.0843 

0.9964 

10 

5 0 

0.0872 

0.9962 

O 85 

10 

0.0901 

0.9959 

50 

20 

0.0929 

0.9957 

40 

30 

0.0958 

0.9954 

30 

40 

0.0987 

0 9951 

20 

50 

0.1016 

0 9948 

10 

6 0 

0.1045 

0.9945 

O 84 

10 

0.1074 

0.9942 

50 

20 

0.1103 

0.9939 

40 

30 

0.1132 

0.9936 

30 

40 

0.1161 

0.9932 

20 

50 

0.1190 

0.9929 

10 

7 0 

0.1219 

0.9925 

O 83 

10 

0.1248 

0.9922 

50 

20 

0.1276 

0 9918 

40 

30 

0.1305 • 

0.9914 

30 

© / 

Cos. 

Sin. 

/ O 


o / 

Sin. 

Cos. 


7 30 

0.1305 

0.9914 

30 

40 

0.1334 

0.9911 

20 

50 

0.1363 

0.9907 

10 

8 O 

0.1392 

0.9903 

O 82 

10 

0.1421 

0.989$) 

50 

20 

0.1449 

0.9894 

40 

30 

0.1478 

0.9890 

30 

40 

0.1507 

0.9886 

20 

50 

0.1536 

0.9881 

10 

9 O 

0.1564 

0.9877 

O 81 

10 

0.1593 

0.9872 

50 

20 

0.1622 

0.9868 

40 

30 

0.1650 

0.9863 

30 

40 

0.1679 

0.9858 

20 

50 

0.1708 

0.9853 

10 

10 0 

0.1736 

0.9848 

O 80 

10 

0.1765 

0.9843 

50 

20 

0.1794 

0.9838 

40 

30 

0.1822 

0.9833 

30 

40 

0.1851 

0.9827 

20 

50 

0.1880 

0.9822 

10 

11 O 

0.1908 

0.9816 

O 79 

10 

0.1937 

0.9811 

50 

20 

0.1965 

0.9805 

40 

30 

0.1994 

0.9799 

30 

40 

0.2022 

0.9793 

20 

50 

0.2051 

0.9787 

10 

12 0 

0.2079 

0.9781 

O 78 

10 

0.2108 

0.9775 

50 

20 

0.2136 

0.9769 

40 

30 

0 2164 

0.9763 

30 

40 

0.2193 

0.9757 

20 

50 

0.2221 

0.9750 

10 

13 O 

0.2250 

0.9744 

O 77 

10 

0.2278 

0.9737 

50 

20 

0.2306 

0.9130 

40 

30 

0.2334 

0.9724 

30 

40 

0.2363 

0.9717 

20 

50 

0.2391 

0.9710 

10 

14 O 

0.2419 

0.9703 

O 76 

10 

0.2447 

0.9696 

50 

20 

0 2476 

0.9689 

40 

30 

0.2504 

0.9681 

30 

40 

0.2532 

0.9674 

20 

50 

0.2560 

0.9667 

10 

15 O 

0.2588 

0.9659 

O 75 


Cos. 

Sin. 

/ o 


82° 30 —90 


75°—82° 30 





















































































157 


TABLE VII.—NATURAL SINES AND COSINES-(Con«m£ed.) 

15 —22° 30' 22° 30'—30° 


o / 

Sin. 

Cos. 


15 0 

0.2588 

0.9659 

0 75 

10 

0.2616 

0.9652 

50 

20 

0.2644 

0.9644 

40 

30 

0.2672 

0.9636 

30 

40 

0.2700 

0.9628 

20 

50 

0.2728 

0.9621 

10 

16 0 

0.2756 

0.9613 

O 74 

10 

0.2784 

0.9605 

50 

20 

0.2812 

0.9596 

40 

30 

0.2840 

0.9588 

30 

40 

0.2868 

0.9580 

20 

50 

0.2896 

0.9572 

10 

17 O 

0.2924 

0.9563 

O 73 

10 

0.2952 

0.9555 

50 

20 

0.2979 

0.9546 

40 

30 

0.3007 

0.9537 

30 

40 

0.3035 

0.9528 

20 

50 

0.3062 

0.9520 

10 

18 0 

0.3090 

0.9511 

O 72 

10 

0.3118 

0.9502 

50 

20 

0.3145 

0.9492 

40 

30 

0.3173 

0.9483 

30 

40 

0.3201 

0.9474 

20 

50 

0.3228 

0.9465 

10 

19 0 

0.3256 

0.9455 

O 71 

10 

0.3283 

0.9446 

50 

20 

0.3311 

0.9436 

40 

30 

0.3338 

0.9426 

30 

40 

0.3365 

0 9417 

20 

50 

0.3393 

0.9407 

10 

20 0 

0 3420 

0.9397 

O 70 

lo 

0.3448 

0.9387 

50 

20 

0.3475 

0.9377 

40 

30 

0.3502 

0.9367 

30 

40 

0.3529 

0.9356 

20 

50 

0.3557 

0.9346 

10 

21 0 

0.3584 

0.9336 

O 69^ 

10 

0.3611 

0.9325 

50 

20 

0.3638 

0.9315 

40 

30 

0.3665 

0.9304 

30 

40 

0.3692 

0.9293 

20 

50 

0.3719 

0.9283 

10 

22 O 

0.3746 

0.92 72 

O 68 

10 

0.3773 

0.9261 

50 

20 

0.3800 

0.9250 

40 

30 

0.3827 

0.9239 

30 67 


Cos. 

sp: 

Sin. 

/ O 


o / 

Sin. 

Cos. 


22 30 

0.3827 

0.9239 

30 67 

40 

0.8854 

0.9228 

20 

50 

0.3881 

0.9216 

10 

23 O 

0.3907 

0.9205 

O 67 

10 

0.3934 

0.9194 

50 

20 

0.3961 

0.9182 

40 

30 

0.3987 

0.9171 

30 

40 

0.4014 

0.9159 

20 

50 

0.4041 

0.9147 

10 

24 O 

0.4067 

0.9135 

O 66 

10 

O’. 4094 

0.9124 

50 

20 

0.4120 

0.9112 

40 

30 

0.4147 

0.9100 

30 

40 

0.4173 

0.9088 

20 

50 

0.4200 

0.9075 

10 

25 O 

0.4226 

0.9063 

O 65 

10 

0.4253 

U.9051 

50 

20 

0.4279 

0.9038 

40 • 

30 

0.4305 

0.9026 

30 

40 

0.4331 

0.9013 

20 

50 

0.4358 

0.9001 

10 

26 O 

0.4384 

0.8988 

O 64 

10 

0.4410 

0.8975 

50 

20 

0.4436 

0.8962 

40 

30 

0.4462 

0.8949 

30 

40 

0.4488 

0.8936 

20 

50 

0.4514 

0.8923 

10 

27 O 

0.4540 

0.8910 

O 63 

10 

0.4566 

0 8897 

50 

20 

0.4592 

0.8884 

40 

30 

0.4617 

0.8870 

30 

40 

0.4643 

0.8857 

20 

50 

0.4669 

0.8843 

10 

28 O 

0.4695 

0.8829 

O 62 

10 

0.4720 

0.8816 

50 

20 

0.4746 

0.8802 

40 

30 

0.4772 

0.8788 

30 

40 

0.4797 

0.8774 

20 

50 

0.4823 

0.8760 

10 

29 O 

0.4848 

0.8746 

O 61 

10 

0.4874 

0 8732 

50 

20 

0.4 Q 99 

0.8718 

40 

30 

0.4924 

0.8704 

30 

40 

0.4950 

0.8689 

20 

50 

0.4975 

0.8675 

10 

30 O 

0.5000 

0.8660 

O 60 


Cos. 

Sin. 

/ O 

i 


67° 30—75 


60°—67° 30 




























































































158 


TABLE VII—NATURAL SINES AND COSINES— (Continued.) 

30°—37° 30' 37° 30 —45° 


o / 

Sin. 

Cos. 


30 O 

0.5000 

0.8660 

O 60 

10 

0.5025 

0.8646 

50 

20 

0.5050 

0.8631 

40 

30 

0.5075 

0.8616 

30 

40 

0.5100 

0.8601 

20 

50 

0.5125 

0.8587 

10 

31 O 

0.5150 

0.85 72 

O 59 

10 

0.51.j 

0.8557 

50 

20 

0-5200 

0.8542 

40 

30 

0.5225 

0.8526 

30 

40 

0.5250 

0.8511 

20 

50 

0.5275 

0.8496 

10 

32 O 

0.5299 

0.8480 

O 58 

10 

0.5324 

0.8465 

50 

20 

0.5348 

0.8450 

40 

30 

0.5373 

0.8434 

30 

40 

0.5398 

0.8418 

•20 

• 50 

0.5422 

0.8403 

10 

33 O 

0.5446 

0.8387 

O 57 

10 

0.54.1 

0.8371 

50 

20 

0.5495 

0.8355 

40 

30 

0.5519 

0.8339 

30 

40 

0.5544 

0.8323 

20 

50 

0.5508 

0.8307 

10 

34 O 

0.5592 

0.8290 

O 56 

10 

0.5010 

0.8274 

50 

20 

0.5640 

0.8258 

40 

30 

0.5664 

0.8241 

30 

40 

0.5688 

0.8225 

20 

50 

0.5712 

0.8208 

10 

35 O 

0.5736 

0.8192 

O 55 

10 

0.5760 

0.8175 

50 

20 

0.5783 

0.8158 

40 

30 

0.5807 

0.8141 

30 

40 

0.5831 

0.8134 

20 

50 

0.5854 

0.8107 

10 

36 O 

0.5878 

0.809t#. 

O 54 

10 

0.5901 

0.8073 

50 

20 

0.5925 

0.8056 

40 

30 

0.5948 

(*8039 

30 

40 

0.5972 

0.8021 

20 

50 

0,5995 

0.8004 

10 

37 O 

0.6018 

0.7986 

O 53 

10 

0.6011 

0.7969 

50 

20 

0.6005 

0.7951 

40 

30 

0.6088 

0.7934 

30 52 


Cos. 

Sin. 

/ O 


o / 

Sin. 

Cos. 

37 30 

0.6088 

0.7934 

40 

0.6111 

0.7916 

50 

0.6134 

0.7898 

38 O 

0.6157 

0.7830 

10 

0.6180 

0.7862 

20 

0.6202 

0.7844 

30 

0.6225 

0.7826 

40 

0.6248 

0.7808 

50 

0.6271 

0.7790 

39 O 

0.6293 

0.7771 

10 

0.6316 

0.7753 

20 

0.6338 

0.7735 

30 

0.6361 

0.7716 

40 

0.6383 

0.7698 

50 

0.6406 

0.7679 

40 O 

0.6428 

0.7660 

10 

0.6450 

0.7612 

20 

0.6472 

0.7623 

30 

0.6494 

0.7604 

40 

0.6517 

0.7585 

50 

0.6539 

0.7566 

41 O 

0.6561 

0.7547 

10 

0.6583 

0.7528 

20 

0.6604 

0.7509 

30 

0.6626 

0.7490 

40 

0.6648 

0.7470 

50 

0.6670 

0.7451 

42 O 

0.6691 

0.7431 

10 

0.6713 

0.7412 

20 

0.6734 

0.7392 

30 

0.6756 

0.7373 

40 

0.6777 

0.7353 

50 

0.6799 

0.7333 

43 O 

0.6820 

0.7314 

10 

0.6841 

0.7294 

20 

0.6862 

0.7274 

30 

0.6884 

0.7254 

40 

0.6905 

0.7234 

50 

0.6926 

0.7214 

44 O 

0.6947 

0.7193 

10 

0.6967 

0.7173 

20 

0.6988 

0.7153 

30 

0.7009 

0.7133 

40 

0.7030 

0.7112 

50 

0.7050 

0.7092 

45 O 

0.7071 

0.7071 


Cos. 

Sin. 


30 5:2 
20 
10 

0 52 

50 

40 

30 

20 

10 

O 51 

50 

40 

30 

20 

10 

O 50 

50 

40 

30 

20 . 

10 

O 49 

50 

40 

30 

20 

10 

O 48 

50 

40 

30 

20 

10 

O 47 

50 

40 

30 

20 

10 

O 46 

50 

40 

30 

20 

10 

O 45 



52° 30-60° 


45°—52° 30' 


































































































159 


TABLE VIII.—MEAN REFRACTIONS. [Bessel.] 
[True for barometer at 29".6, temperature at 48° F.] 


Alt. 

Refr. 

Alt. 

Refr. 

Alt. 

Refr. 

' Alt. 

Refr. 

0° O' 

34' 

54" 

3° 

O' 

14' 

15" 

10° 

5' 

16" 

24° 

2' 

09" 

10 

32 

49 


30 

12 

48 

11 

4 

48 

26 

1 

58 

20 

30 

52 

4 

0 

11 

39 

12 

4 

25 

28 

1 

48 

30 

29 

03 


30 

10 

40 

13 

4 

05 

30 

1 

40 

40 

27 

23 

5 

0 

9 

46 

14 

3 

47 

35 

1 

22 

50 

25 

50 


30 

9 

02 * 

15 

3 

32 

40 

1 

09 

1 0 

24 

25 

6 

0 

8 

23 

16 

3 

19 

45 

0 

58 

10 

23 

07 


30 

7 

49 

17 

3 

07 

50 

0 

48 | 

20 

21 

56 

7 

0 

7 

20 

18 

2 

56 

60 

0 

33 

30 

20 

51 


30 

6 

53 

19 

2 

46 

70 

0 

21 

40 

19 

52 

8 

0 

6 

30 

20 

2 

37 

80 

0 

10 

50 

18 

58 


30 

6 

08 

21 

2 

29 

90 

0 

0 

2 0 

18 

09 

9 

0 

5 

49 

22 

2 

22 




30 

16 

01 


30 

5 

32 

23 

2 

15 





FACTOR B. FACTOR t. FACTOR T. 


Temp. 

External 

Air. 

T 

Temp.. 
Externa 1 
Air. 

T 

O 

o 

1 

1.156 

40° F. 

1.017 

- 10 

1.130 

50 

.998 

0 

1.106 

60 

.978 

+ 10 

1.082 

70 

.960 

20 

1.060 

80 

.942 

30 

1.038 

90 

.925 


Attached 

Therm. 

t 

- 20° F. 

1.005 

0 

1.003 

+ 20 

1.001 

+ 40 

.999 

-f 60 

.997 

+ 80 

.996 


Barom. 

B 

28' 

.0 

0.946 

28 

.5 

0.963 

20 

.0 

0.980 

29 

.5 

0.997 

30 

.0 

1.014 

30 

.5 

1.031 

31 

.0 

1.047 


True refraction = mean refraction X B X t X T. 












































160 TABLE IX.— EFFECT OF REFRACTION ON DECLINATION. 


O cj 


Declination. 


23° 27' 


"I - 

20 ° 


+ 

15° 


+ 

10 ° 


4 

5° 


0 ° 


10 ° 


15° 


20° 23° 27' 


/ 

— 4 

— 3 

— 1 

+ 4 

13 
0 34 


0 

0 

+ 3 
7 
17 
3 39 


+ 5 
6 
8 
12 
22 
0 47 


—j-0 10 
11 
13 
18 
28 
0 5 1 


4-0 15 
16 
18 
23 
0 35 
1 07 


+0 21 
22 
24 
30 
0 42 
1 19 


-f 0 27 
28 
30 
36 
0 50 
1 36 


2 u6 


3 l; 


4 24 


6 43 


11 56 


0 07 
08 
10 
16 
28 
0 53 
2 11 


0 10 
11 
14 

20 
32 
0 59 


0 15 
16 
19 
26 
0 39 
1 10 


12 33 


3 23 


0 21 
22 
25 
32 
0 47 
1 24 


0 27 
28 
32 
40 
0 56 
1 43 


0 33 
35 
39 
0 47 
1 06 
2 06 


0 40 
42 
46 
0 56 

1 19 

2 41 


0 33 
34 
37 
0 44 

i oo! 

1 57 


0 40 
41 
45 
0 52 
1 11 


0 48 
50 
0 55 
1 06 
1 36 


3 36 


4 51 


8 00 


28 


0 58 
0 59 
1 06 
1 19 

1 58 


5 18 


0 48 
49 
0 54 
1 02 
1 25 


3 14 


1 09 
1 11 
1 18 

1 35 

2 28 


8 54 


0 55 
0 56 
1 00 
1 11 
1 36 


4 01 


1 18 
1 20 
1 27 

1 51 

2 59 


16 42 


0 17 
18 
22 
29 
0 41 

1 07 

2 11 


0 21 
22 
26 
33 
0 47 
1 16 
2 35 


0 27 
28 
32 
41 
0 56 
1 31 


0 33 
34 
39 
0 48 
1 07 
1 51 


0 40 
42 
47 
0 58 
1 20 
2 20 


0 48 
50 
0 56 
1 09 
1 36 
3 02 


0 58 
1 00 
1 07 
1 22 
1 58 
| 4 15 


1 09 
1 11 
1 19 

1 39 

2 30 


1 22 
1 25 

1 36 

2 02 
3 19 


1 40 
1 43 

1 58 

2 36 


4 41 


6 47 


14 18 


3 27 5 u3 


8 4; 


22 26 


1 55 

2 00 
2 18 
3 09 


6 45 


0 29 
30 
34 
42 
0 55 
1 20 
2 11 


0 33 
34 
39 
0 48 


0 48 
50 
0 56 
1 07 
1 29 
220 
30 | 5 09 


0 58 
1 00 
1 07 
1 21 
1 49 
3 02 


1 09 
1 11 
1 19 

1 37 

2 17 


1 22 
1 25 
1 36 

1 59 

2 58 


1 40 
1 44 

1 58 

2 32 
4 07 


y 12 


| 4 16 

26 44 


6 52 


15 08 


2 03 
2 09 

2 29 

3 22 
6 32 


2 37 

2 46 

3 18 


| 4 55 
13 37 


3 12 

3 24 

4 13 


7 u; 


0 43 
44 
48 
0 56 
1 10 

1 32 

2 12 


0 48 
50 
0 54 


0 58 
0 59 


2 37 


1 09 
1 11 
1 18 
1 31 

1 57 

2 50 


1 22 
1 25 
1 34 

1 51 

2 28 
3 54 


1 40 
1 43 

1 55 

2 21 
3 17 


2 03 
2 08 
2 26 

3 04 

4 44 


2 37 

2 44 

3 11 
419 

| 8 08 


3 32 

3 46 

4 34 
| 7 05 
22 10 


| 5 12 


9 30 


| 6 21 

30 13 


12 17 


5 16 
|T44 
| 7 44 
16 40 


7 46 

8 45 
13 44 


O) 

■1 bD 
3 G 
O as 


4 " 

23° 27' 


+ 

20 ° 


+ 

15° 


+ 

10 ° 


4 


0 ° 


5° 


10 ° 


15° 


20 ° 


23° 27' 


Declination. 


















































































table x.-errors in azimuth.* 


161 


— due to error 
of 1' in 


Declination 


Latitude 




ALTITUDE 


Hour-angle. 


1 

2 

3 

4 

5 

6 


1 

2 

3 

4 

5 

6 


Latitude — 


+ 20° 

+ 30° 

-f 40° 

-f 50° 

4 60° 

4M1 

4'. 46 

5'. 04 

6'. 01 

7'. 73 

2 .13 

2 .31 

2 .61 

3 .11 

4 .00 

1 .50 

1 .63 

1 .85 

2 .20 

2 .83 

1 .23 

1 .33 

1 .51 

1 .80 

2 .31 

1 .10 

1 .20 

1 .35 

1 .61 

2 .07 

1 .06 

1 .15 

1 .31 

1 .56 

2 .00 

3'. 97 

4'.31 

4'. 87 

5'. 81 

7'. 46 

1 .84 

2 .00 

2 .26 

2 .69 

3 .46 

1 .06 

1 .15 

1 .31 

1 .56 

2 .00 

0 .61 

0 .67 

0 .75 

0 .90 

1 .15 

0 .29 

0 .31 

0 .35 

0 .42 

0 .54 

0 .00 

0 .00 

0 .00 

0 .00 

0 .00 


Lati¬ 

tude. 

Hour- 

angle. 


Declination - 

8 . 

+ 23° 27' 

4-10° 

0° 

— 10° 


1 

O'. 85 

2'. 44 

3'. 35 

3'. 69 


2 

0 .05 

0 .82 

1 .25 

1 .52 

20° 

3 

0 .11 

0 .48 

0 .69 

0 .85 


4 

0 .19 

0 .37 

0 .48 

0 .56 


5 

0 .26 

0 .34 

0 .39 

0 .42 


6 

0 .34 

0 .36 

0 .36 



1 

2 .19 

3 .75 

4 .07 

4 .21 


2 

0 .80 

1 .48 

1 .75 

1 .90 


3 

0 .54 

0 .87 

1 .03 

1 .14 

30° 

4 

0 .48 

0 .65 

0 .74 

0 .80 


5 

0 .48 

0 .57 

0 .61 

0 .64 


6 

0 .54 

0 .57 

0 .58 



1 

4 .19 

4 .69 

4 .82 

4 .88 


2 

1 .69 

2 .10 

2 .24 

2 .32 


3 

1 .05 

1 .28 

1 .41 

1 .48 

40° 

4 

0 .83 

0 .98 

1 .05 

1 .09 


5 

0 .77 

0 .85 

0 .89 

0 .91 


6 

0 .80 

0 .83 

0 .84 



1 

5 .64 

5 .82 

5 .88 

5 .90 


2 

2 .59 

2 .80 

2 .87 

2 .91 


3 

1 .66 

1 .83 

1 .89 

1 .93 

50° 

4 

1 .29 

1 .40 

1 .45 

1 .48 


5 

1 .15 

1 .22 

1 .25 

1 .26 


6 

1 .15 

1 .18 

1 .-19 



1 

7 .56 

7 .62 

7 .64 

7 .65 


2 

3 .71 

3 .81 

3 .84 

3 .86 


3 

2 .49 

2 .58 

2 .62 

2 .64 

60° 

4 

1 .96 

2 .04 

2 .07 

2 .08 


5 

1 .74 

1 .79 

1 .81 



6 

1 .69 

1 .73 

1 .73 



23° 27' 


3'. 96 
1 .71 
0 .99 
0 .64 
0 .45 


4 .29 
2 .00 
1 .23 
0 .86 
0 .65 


4 .92 
2 .38 
1 .53 
1 .12 


5 .92 
2 .94 
1 .96 


7 .67 
3 .87 


* For method of using this table see pp. 94, 95. 




















































































162 


TABLE XI.—VALUES OF USED IN THE REDUCTION TO THE 

MERIDIAN FOR CIRCUMMERIDIAN ALTITUDES. 



0 sec. 

10 sec. 

20 sec. 

30 sec. 

40 sec. 

50 sec. 

Min. 

0 

0".0 

0'M 

0".2 

0".5 

0".9 

1".4 

1 

2 

3 

3 

4 

5 

7 

2 

8 

9 

11 

12 

14 

. 16 

3 

18 

20 

22 

24 

26 

29 

4 

31 

34 

37 

40 

43 

46 

5 

49 

.52 

56 

59 

63 

67 

6 

71 

75 

79 

83 

87 

92 

7 

96 

101 

106 

110 

115 

120 

8 

126 

131 

136 

142 

147 

153 

9 

159 

165 

171 

177 

183 

190 

10 

196 

203 

210 

216 

223 

230 

11 

238 

245 

252 

260 

267 

275 

12 

283 

291 

299 

307 

315 

323 

13 

332 

340 

349 

358 

367 

376 

14 

385 

394 

403 

413 

422 

432 

15 

442 

451 

461 

472 

482 

492 

16 

502 

513 

524 

534 

545 

556 

17 

567 

578 

590 

601 

612 

624 

18 

636 

648 

660 

672 

684 

696 

19 

708 

721 

733 

746 

759 

772 

20 

785 

798 

811 

825 

838 

852 


TABLE XII—APPARENT DIP OF THE HORIZON. 


Height 
of , 
Sex¬ 
tant. 

Dip. 

Heigh' 

of 

Sex¬ 

tant. 

Dip. 

Height 

of 

Sex¬ 

tant. 

Dip. 

Feet. 

/ 

n 

Feet. 

f 

n 

Feet. 

/ n 

6 

2 

24 

11 

3 

15 

16 

3 55 

7 

2 

36 

12 

3 

24 

17 

4 03 

8 

2 

46 

13 

3 

32 

18 

4 10 

9 

2 

56 

14 

3 

40 

19 

4 16 

10 

3 

06 

15 

3 

48 

20 

4 23 


Height 

of 

Sex- 
j tant. 

Dip. 

Height 

of 

Sex¬ 

tant. 

Dip. 

Feet. 

/ 

// 

Feet. 

/ 

// 

21 

4 

30 

26 

5 

0 

22 

4 

36 

27 

5 

06 

23 

4 

42 

28 

5 

11 

24 

4 

48 

29 

5 

17 

25 

4 

54 

30 

5 

22 


















































•• • 




















. 





















































































» 











































